successive estimations of bilateral bounds and trapping
TRANSCRIPT
1
Successive Estimations of Bilateral Bounds and Trapping/Stability Regions of Solution to Some Nonlinear
Nonautonomous Systems
Mark A. Pinsky
Abstract. Estimation of the degree of stability and the bounds of solutions to nonautonomous nonlinear systems
present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative
conditions which are unable to effectively gage these characteristics in time-varying nonlinear systems. This paper
develops a novel methodology providing successive approximations to solutions that are stemmed from the
trapping/stability regions of these systems and estimate the errors of such approximations. In turn, this leads to
successive approximations of both the bilateral bounds of solutions and the boundaries of trapping/stability regions
of the underlying systems. Along these lines we formulate enhanced stability/boundedness criteria and contrast our
inferences with inclusive simulations which reveal dependence of the trapping/stability regions upon the structure of
time – dependent components and initial time- moment.
Keywords: Stability of nonautonomous nonlinear systems; Bilateral Bounds of Solutions; Estimation of
trapping/stability regions.
1. Introduction
Stability and boundedness of solutions to nonlinear systems with variable coefficients are of major concerns in
various applications. For instance, the corresponding stability conditions aid the design of robust controllers and
observers [1-3]. Sufficient conditions of stability of fixed solutions to such systems were initially established by
Lyapunov [4] and subsequently advanced in various publications, see [5-7] and review paper [8] for additional
references. In control literature, the method of Lyapunov functions has been used extensively for stability analysis of
some nonlinear and nonautonomous systems [9-15]. Yet, adequate Lyapunov functions are rare in this field.
Convenient conditions embracing boundedness/stability of solutions to nonhomogeneous nonlinear systems with
variable coefficients are practically unknown as well.
Nonetheless, local stability conditions are well known for nonlinear systems with time – invariant and Hurwitz
linearization and smooth or Lipschitz continuous nonlinear perturbations [1, 2, 5]. Such conditions can be naturally
extended for nonlinear systems with periodic linearization. Yet, local dynamics of systems with nonperiodic
linearization turn out to be more complex and sensitive to small perturbations. Consequently, assessment of local
stability of such systems requires specialized techniques resting on application of Lyapunov and generalized
exponents [4, 5].
Let us briefly review some explicit stability conditions for nonautonomous nonlinear systems which can be
written as follows,
( ) ( ) ( ) ( )
( )
0
0 0 0 0
, , , , ,0 0
, ,
nx A t x f t x F t t t x f t
x t t x x
= + + =
= (1.1)
where, 0: [ , ) n nf t= → and
0: [ , ) nF t= → are continuous functions, matrix, ( ) n nA t is continuous,
0t t , 0 : [ , ),t = , 0
nx , is a bounded neighborhood about 0x , ( ) ( )0F t F t= ,
( )0
0 0sup 1,t t
t F
= , 0 is a set of all nonnegative real numbers, stands for induced 2-norm of a matrix
or 2-norm of a vector and ( )0 0 0, , : [ , ) n nx t t x t= → is the solution to (1.1). To shorten the notation,
we will write below that ( ) ( )0 0 0, , ,x t t x x t x and assume that (1.1) possesses a unique solution 0t t .
We will also reference the homogeneous counterpart of (1.1),
( ) ( )
( )0 0 0
,
,
x A t x f t x
x t x x
= +
= (1.2)
and its linearization,
( )
( )0 0 0,
x A t x
x t x x
=
= (1.3)
2
Obviously, the solution to (1.3) can be written as ( ) ( )0 0 0, ,x t x W t t x= ,where ( ) ( ) ( )1
0 0 0 0, , ,W t t w t t w t t−=
and ( )0,w t t are transition and fundamental solution matrices for (1.3), respectively.
_________________________________________________________ Mark A. Pinsky. Department of Mathematics and Statistics, University of Nevada.Reno, Reno NV 89557, USA. e-mial:[email protected]. To appear in Journal of Nonlinear Dynamics.
Contemporary stability conditions of the trivial solution to (1.2) are frequently derived under Lipschitz continuity
condition,
( ) ( )1 0 1 0, , , ,nf t x l t t x x t t (1.4)
(where 1 is a bounded neighborhood of 0x and ( )1 0 1 0 0ˆ, ,l t t l t t is a continuous function) and some
bounds on the norm of transition matrix [5- 8],
( ) 0( )
0 0, , , , 0v t t
W t t Ne t t N v− −
(1.5)
or
0
0 0( , exp ( ) , 0
t
t
W t t N s ds t t (1.6)
Then, asymptotic stability of the trivial solution to (1.2) is embraced if either (1.4), (1.5) and
1ˆ 0Nl v− (1.7)
hold [5, 6] or (1.4), (1.6) and
( ) ( )0
0 1ˆlimsup 1/ ( ) 0
t
tt
t t s ds Nl→
− + (1.8)
hold [7, 8]. Clearly, (1.6) reduces to (1.5) if ( )t const = , otherwise, to our knowledge, this function was not explicitly
defined yet for systems of practical concerns. Utility of Lipschitz constant, i.e. 1l̂ , in (1.7) and (1.8) usually contributes
to overconservative nature of these conditions [16, 17]. Additionally, efficient bounding of ( )0,w t t can present a
challenging task even if ( ) ( )0 0, expw t t A t t= − with A const= [18]. More efficient stability conditions of trivial
solution to (1. 2) were developed in [19], but verification of these conditions can be challenging by its own.
Virtually all known stability conditions for nonlinear time-varying systems turn out to be overconservative and
local in nature - they unable to adequately estimate the region of stability for these systems. However, the problem
of estimating the stability regions for autonomous nonlinear systems has attracted a flood of publications in the past
few decades, see recent reviews in [20,21]. Most of the efforts in this area are based on optimizing the Lyapunov
functions and combining this methodology with the level set method [20 – 28]. Another popular tactic is trajectory
reversing [29 – 33] which is also combined with level set method [34, 35]. There are also other techniques
exploring geometrical reasoning [36-38] and some other strategies [39]. Yet, neither of these techniques were
targeted nonautonomous nonlinear systems at the present time.
So-known geometric methods were developed to stability analysis of nonlinear and nonlinear controlled systems,
see, e.g., [2, 3, 49] and subsequent references therein.
Though stability assesses the fate of solutions on long time – intervals, estimation of their evolutions on
initial/intermediate time – intervals is frequently equally imperative [40 – 42]. Nonetheless, this problem has not
received sufficient attention in the contemporary literature.
In [43] under some conditions we develop an approach to estimation of the upper bound of norms of solutions to
(1.1)/(1.2) which leads to more general boundedness/ stability criteria and estimation of the trapping/stability
regions for these systems. Still, the developed estimates turn out to be rather conservative, especially for
nonhomogeneous systems and ones with large nonlinear components.
This paper develops a novel methodology which successively approximates the solutions stemming from
trapping/stability regions of systems (1.1)/(1.2) and estimate the norms of errors of these approximations. This leads
to recursive estimation of both the bilateral bounds of solutions’ norms and the trapping/stability regions. Along
these lines we formulate novel bondedness/stability criteria and reinforce our theoretical results in representative
simulations. Our inferences advance the methodology developed in [43] and uplift some of its major limitations.
This paper represents a solution of (1.1) as,
3
( ) ( ) ( )0 0 0 0, , , , 1 ,mx t x Y t x z t x m t t= + (1.9)
where 0: [ , ) n n
mY t= → , ( ) ( )0 0 00, 0,mx x Y x x= = , ( )00, 0z x = and is a set of positive integers.
Thus, ( )0,mY t x can be viewed as an approximate solution to (1.1) assuming its initial data and ( )z t is considered
as the error-function of this approximations that depends implicitly upon 0x . Clearly, (1.9) yields the following
bilateral bounds,
( ) ( ) ( ) ( ) ( )0 0 0 0 0 0, , , , , ,m mY t x z t x x t x Y t x z t x t t− + (1.10)
where the left bound should be adjusted to zero if it takes negative values.
Next, we represent ( )0,mY t x in the following form,
( ) ( )0 0
1
, ,m
m i
i
Y t x y t x=
= (1.11)
where0: [ , ) n n
iy t= → are solutions to some linear equations which are derived subsequently.
Afterward, we show under some conditions that either ( )0lim , 0mt
Y t x→
= if 0 0F = and 0x belongs to the
stability region of the trivial solution of (1.2) or ( ) ( )0 0lim ,mt
Y t x O F→
if 0 0F and 0x belongs to the
trapping region of (1.1). Thus, in both cases the behavior of ( )0,x t x on long time intervals is determined by
evolution of ( )0,z t x .
This paper is organized as follows. Next section presents the synopsis of some essential technical results. Sections
3 and 4 derive equations for ( )0,iy t x and ( )0,z t x for a relatively simple system and, subsequently, in a general
form. Section 5 examines behavior of the approximate solutions on long time – intervals, section 6 derives a linear
scalar equation for the upper bond of ( )0,z t x and develops its applications, section 7 derives a nonlinear
counterpart of this equation and apply it for estimating of the bilateral bounds of solutions and the trapping/stability
regions. Section 8 review the results of our simulations and section 9 concludes this study and outlines the topics of
subsequent works. The appendix derives some technical results pertained to section 7.
2. Preliminaries
2.1. Lyapunov and General Exponents
Firstly, let us briefly contrast the definitions of broadly used the Lyapunov exponents, see, e.g. [8] and the general
exponents, which were introduced in [5]. The latter frequently provide more conservative inferences but possess
“more natural properties”. In fact, the spectrum of Lyapunov exponents for (1.3) sensitively depends upon
perturbations of the underlying system [44-46] whereas the maximal/minimal general exponents are robust under
such perturbations [5] which enables our inferences in section 5. Different terminology and notations are currently
used for these measures and we adopt ones from [8] and [5], respectively.
For equation (1.2) the Lyapunov exponents, ( )( ) ( ) ( )0
1
0 00 lim sup ln , ,,it t
t t x t x i nx t x−
→= − , where ( )0,x t x is
a solution to (1.2). The maximal Lyapunov exponent, ( )( ) 0max ,ii
x t x = measures the maximal rate of
exponential growth/decay of the corresponding solutions as t→and bears a pivoting role in stability theory. For
a linear system (1.3) the Lyapunov exponents are defined as, ( ) ( )( )0
1
0lim sup ln ,i it t
t t w i nt −
→= − , where i
are the singular values of the fundamental matrix of (1.3); and for linear systems, max ii
= [8]. This let to bound
the transition matrix of (1.3) as follows,
4
( ) ( )( )( )0 0 0, exp , 0,W t t D t t D t t + − (2.1)
where is a small positive number [5].
In turn, the upper general exponent, , is defined in [5] as follows. Let ( )0,x t x with 0x r be a solution
of (1.2) stemming from a centered at zero ball with radius r . Assume that every of such solutions obeys the
inequality
( ) ( )( ) ( )0 0 0 0, exp , , 0,x t x N v t t x t N v t t − − (2.2)
Then, infimum of values of v = − , for which (2.2) holds, is called the upper general exponent for (1.2) at zero.
Clearly, supv− = for which (2.1) can be defined. It is stated in [5] that as well as that the trivial
solution to (1.2) is exponentially stable if 0 .
For linear equation (1.3) condition 0x r is dropped from the definition of A and for this equation (2.2)
implies that,
( ) ( )( )0 0 0, exp , , 0,W t t N v t t N v t t − − (2.3)
Note also that (2.3) is the necessary and sufficient condition for asymptotic stability of (1.3) [2].
In section 5 we use some of the theorems from sections 3.4 and 3.5 in [5] which are abbreviated below.
Statement 1. Consider a linear equation,
( ) ( )( ) , nx A t B t x x= + (2.4)
where both, ( )A t and ( ) 0,n nB t t t are continuous matrices. Assume that,
( )1
,lim sup
t s
t st
s B d +
−
→ (2.5)
Then, 0 there is such ( ) that, A B A + + , where A and A B + are maximal general exponents for
equations (1.3) and (2.4), respectively. Note that (2.5) is contented if, e.g., ( ) * 0,B t t t t .
Furthermore, it follows from the above statement that if 0A and is sufficiently small, then, ( ) 0A B
+
as
well.
Statement 2. Assume that 0 = . Then, A B A + = and
( ) ( )( )0 * 0 * 0, exp , 0,A B AW t t N t t N t t+ − , where ( )0,A BW t t+ is a transition matrix of (2.4). The
last conditions are intact if, e.g. ( )lim 0t
B t→
= .
2.2. Comparison Principle
This paper references the comparison principle [2] which is outlined as follows for convenience. Let us consider
the IVP for a scalar differential equation,
( ) ( )
( )
0
0 0
, , ,x g t x t t x t
x t x
=
= (2.6)
where function ( ),g t x is continuous in t and locally Lipschitz in x for x . Suppose that the solution to
(2.6), ( )0 † 0, ,x t x X t t . Next, consider a differential inequality,
( )
( )
0
0 0 0
, ,D y g t y t t
y t y x
+
= (2.7)
5
where D y+ denotes the upper right-hand derivative of solution to (2.6) [2] and ( )0 0, ,y t y t t . Then,
( ) ( )0 0 0, , ,x t x y t y t t .
2.3. Estimation of Solutions’ Norms
Next, we recap our approach for estimating the norm of solutions to (1.1) which is used in sections 6 and 7, see
[43] for additional details. This paper derives a differential inequality for the norms of solutions to (1) in the
following form,
( ) ( ) ( ) ( )
( ) ( )
0 0
1
0 0 0 0 0 0
, , ,
, ,
D x p t t x c t t f t x F t
x t x w t t x X
+
−
+ +
= = (2.8)
where
( ) ( )( )0 0, ln , /p t t d w t t dt= (2.9)
and
( ) ( ) ( ) ( ) ( )1
0 0 0 max min, , , /c t t w t t w t t w w −= = (2.10)
is the running condition number of ( )0,w t t , ( )max t and ( )min t are running maximal and minimal singular
values of ( )w t .
Note also that [43] assumes that ( )0 0, 1w t t = and, hence, (2.9) can be written as follows,
( ) ( )0
0, exp
t
t
w t t p s ds
= (2.11)
Note that below we adopt a shorter notation: ( ) ( ) ( ) ( ) ( ) ( )0 0 0, , , , ,w t t w t p t t p t c t t c t= = = and assume
that ( ) ˆc t c const = .
Due to comparison principle, solutions to (2.8) are bounded from above by solutions of the following scalar
equation,
( ) ( ) ( )( ) ( )
( ) ( )
0
1
0 0 0 0 0
, ,
,
X p t X c f t x t x F tt
X t X w t x X−
= + +
= = (2.12)
Consequently, ( ) ( )0 0 0, , ,X t X x t x t t .
Furthermore, application of (1.4) and standard norm’s inequalities linearizes (2.12) for ( )0 1 0, ,x t x t t
and let us to write this equation as follows,
( ) ( ) ( )
( ) ( )1
0 0 0 0 0,
l l
l
X t X c t F t
X t X w t x X
−
= +
= = (2.13)
where 1p cl = + . Application of (2.13) enabled us to bound the solutions to (1.1) and derived some boundedness
and stability criteria for (1.2). To refine these estimates, we formulated in [43] a nonlinear extension of (1.4), i.e.,
( ) ( ) 2 0, , ,,nf L t x x t tt x (2.14)
where 2 is a bounded neighborhood of 0x and function0 0 0: [ , )L t = → is continuous in t and x ,
( ) 0,0L t = . Furthermore, [43] defined ( ),L t x explicitly if f is either a polynomial in x or can be
approximated by such polynomial with, e.g. Lagrange error term. Let us show for convenience how to define
6
( ),L t x in a simple but typical case. Assume that ( ) ( )2
1 1 2 2 1 2
T
f a t x x a t x x = . Then, 2 1
f f
2
1 1 2a x x 2 1 2a x x+ 3 2
1 2a x a x + .
Thus, (2.14) turns into a global inequality for polynomial vector fields or ones which are approximated by
polynomials with globally bounded error terms.
Application of (2.14) turns (2.8) into the following inequality,
( ) ( ) ( ) ( )( )
( ) ( )
0
1
0 0 0 0 0
,
,
D x p t x c L t x F tt
x t x w t x X
+
−
+ +
= = (2.15)
which, due to comparison principle, yields the equation,
( ) ( ) ( ) ( )( )
( ) ( )
* * *
1
* 0 0 0 0 0
,
,
X p t X c L Ft Xt t
X t X w t x X−
= + +
= = (2.16)
where ( ) ( )* 0 0 0, , ,X t x x t x t t . This last equation frequently enables more accurate estimation of the
bounds of solutions to (1.1), derivation of enhanced stability criteria and estimation of the trapping/stability regions
for equations (1.1) and (1.2), respectively.
Additionally, [43] derives that ( ) ( ) ( )0
0 0 0, exp
t
t
x t x c t p s ds x
, where ( )0,x t x is a solution to (1.3).
Thus, the last formula implies that,
( ) ( )0
0
1
0sup
t
At t
t
v t t p s ds−
= − −
(2.17)
Next, assume that ( ) ( ) ( )0 0 0 0ˆ ˆ, , , 0t t t t t t − . Then, ( ) ( )
0
1
0ˆ
t
t
t t p s ds −
− − since
( ) ( )1 0k t l t and, in turn, ˆv . This last inequality is used in section 6 of this paper.
Finally, we extend some typical definitions of trapping/stability regions used for autonomous systems on
nonautonomous systems (1.1) and (1.2), respectively. Let us firstly acknowledge a set of initial vectors,
( )0 0 0,nx t t , where it is assumed that
( )00 t . Then, we present below the following
definitions.
Definition 1. A compact set of initial vectors, which includes zero-vector, is called a trapping region of equation
(1), if condition, ( )0 0x t implies that ( ) ( )0 0 0 * 0, , ,x t t x t t t t .
Definition 2. An open set of initial vectors, which includes zero-vector, is called a stability region of the trivial
solution to (2) if condition, ( )0 0x t implies that ( )0
0 0lim , , 0t t
x t t x →
= .
We illustrate in section 8 that the trapping and stability regions for nonautonomous systems, in general, depend
upon 0t .
3. Motivating Example
This section demonstrates application of our approach to the design of successive approximation to the norms of
solutions to a Van der-Pol – like model with variable coefficients which was also tackled in [43] by our former
technique. This system can be written as (1.1) and assumes the following form in dimensionless variables,
( )1 1
2 3
22 1 2 2 2
00 1 0x xd
F tx x xdt
= + +
− − − (3.1)
7
where 1 2
Tx x x= ,
3
2 20T
f x = − , ( ) ( )2[0 ]TF Ft t= , ( ) ( )2 2
0 1 0, constt t = + = ,
1 1 1 2 2sin sina rt a r t = + , 2 2 2sin , , , , , 1,2i iF a t a a r const i = = = . Next, we set in (1.11) that1 2
T
m m mY Y Y=
, 1 2 ,T
i i i my y y i= ,where m is a set of m -positive integers, ( )1 0 0y t x= and ( )0 0, 1iy t i= .
Then, substitution of (1.9) and (1.11) into (3.1) returns the following system,
( )
1 1 2 2
1 1
3
2 2 0 1 1 1 2 2 2 2 2
1 1 1 1
m m
i i
i i
m m m m
i i i i
i i i i
y z y z
y z y z y z y z F t
= =
= = = =
+ = +
+ = − + − + − + +
Below we present two natural approaches to resolve this system which embrace the following conditions:
equations for every iy are linear and coupled consecutively such that the prior sets of equations are independent from
subsequent ones. This yields two sets of consecutive linear equations for ,i my i and nonlinear equations for z
which are simulated in section 8.
Assume initially that 1m = and 1x y z= + . This comprises the following systems of linear equations for 1y and
nonlinear equations for z ,
( ) ( )11 21
21 0 11 1 21 1 0 0,
y y
y y y F t y t x
=
= − − + = (3.2)
( ) ( )
1 2
2 2 3 32 0 1 1 2 2 021 2 21 2 2 21
, 03 3
z z
z z z z ty z y z z y
=
= − − − =+ + + (3.3)
Then, application of equation (2.16) to (3.3) yields a scalar equation,
( ) ( ) ( ) ( )2 2 3 3
1 1 2 21 1 21 1 1 21 1 03 3 , 0e e e e e ez p t z c t y z y z z y z t= + + + + = (3.4)
where symbol . stands for absolute value, both ( )p t and ( )c t are defined by (2.9) and (2.10) and ( )w t is a
fundamental matrix of solutions to linearized and homogeneous equation (3.1). Clearly, 1ez z .
For 2m = , 1 2x y y z= + + and the corresponding system of equations includes (3.2) which is appendant by the
following equations,
( ) ( )
12 22
2 2 3
22 12 1 2 21 22 2 21 2 03 , 0
y y
y y y y y y t
=
= − − + − = (3.5)
( )
( )( )22
1 2 0
2 2 3 2
2 22 2 2 21 22 222 0 1 1 2 2
, 0,
3 3 3
z z z t
Y z Y z z y y yz z z
= =
+ + + += − − − (3.6)
where we recall that 22 21 22Y y y= + .
Next, application of (2.16) to (3.6) returns the following equation,
( ) ( ) ( ) ( )22
2 2 3 2
2 2 2 2 22 2 2 21 22 22 2 03 3 3 , 0e e e e e ez p t z c t Y z Y z z y y y z t= + + + + + = (3.7)
where 2ez z .
Finally, for 3m = , the successive approximations include (3.2) and (3.5) which are appendant by the following
equations,
8
( )
( ) ( )22
13 23 3 0
2 2 2
23 13 1 2 23 2 21 22 22
, 0,
3 3
y y y t
y y Y y y y y
= =
= − − + − + (3.8)
( )
( )( )23
1 2 0
2 2 3 2
2 23 2 2 22 23 232 0 1 1 2 2
, 0,
3 3 3
z z z t
Y z Y z z Y y yz z z
= =
+ + + += − − − (3.9)
where 23 21 22 23Y y y y= + + .
Consequently, the upper estimate of the norm of solutions to (3.9) assumes the following equation,
( ) ( ) ( ) ( )2 2 3 2
3 3 2 23 3 23 3 3 22 23 23 3 03 3 3 , 0e e e e e ez p t z c t Y z Y z z Y y y z t= + + + + + = (3.10)
Next, we outline an alternative approach where the underlying matrix of the linear block for all equations in
successive approximations equals ( )A t . To shorten the presentation, we present below only the equations for
3m = that append (3.2) holding in this case,
( )12 22
2 3
22 12 1 22 2 21 2 0, 0
y y
y y y y y t
=
= − − − = (3.11)
( ) ( )
13 23
2 3 3
23 13 1 23 2 22 21 3 0
,
, 0
y y
y y y Y y y t
=
= − − − − = (3.12)
( )
( )1 2 0
2 2 3 3 32 0 1 1 2 2 23 2 23 2 2 23 22
, 0,
3 3
z z z t
z z z Y z Y z z Y Y
= =
= − − − + + + − (3.13)
Afterwards, the upper estimate of the norm of solutions to (3.13) assumes the following equation,
( ) ( ) ( ) ( )2 2 3 3 3
4 4 2 23 4 23 4 4 23 22 4 03 3 , 0e e e e e ez p t z c t Y z Y z z Y Y z t= + + + + − = (3.14)
Hence the transition matrices for equations (3.2), (3.11) and (3.12) are the same which simplifies their numerical
evaluation. Nonetheless, simulations show that for a fixed m the first set of equations is more accurate than the
second one, see section 8.
Clearly, the above procedures can be readily applied to derivation of the underlying equations of our methodology
for system (1) if ( ),f t x is a polynomial in x .
4. Successive Approximations
This section derives the successive approximations in a general form which does not assume that ( ),f t x is a
polynomial in x . Yet, for polynomial systems, the derived equations reduce to ones that can be set up by extending
the prior approach.
Adopting (1.9) and (1.11) for (1) let us to cast equations for ( )0,iy t x and ( )0,z t x into two different forms. In
the first case we assume that for sufficiently small x and 0t t , Jacobean of f in x , i.e., ( ), n nf t x is
a continuous matrix in t and x . Then, the first set of recursive equations for ( )ky t can be written as follows,
9
( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
1 1 1 0 0
2 1 2 1 2 0
3 1 2 3 1 2 1 1 2 3 0
1 1 2 2 1 0
, ,
, , , 0,
, , , , , 0
, , , , , 0,
3
k k k k k k k k
m
y Ay F t y t x
y A f t y y f t y y t
y A f t y y y f t y y f t y f t y y y t
y A f t Y y f t Y f t Y f t Y y y t
k
− − − − −
= + =
= + + =
= + + + + − − =
= + + − − =
(4.1)
( ) ( ) ( ) ( )1 1 0, , , , 0,
2
m m m mz Az f t z Y f t Y f t Y y z t
m
− −= + + − − =
(4.2)
Note that for 1m = , (4.2) takes the following form,
( ) ( )1 0, , 0z Az f t z y z t= + + =
The second set of equations, generalizing our second approach from prior section, can be written as follows,
( ) ( )
( ) ( )
( ) ( ) ( )
1 1 1 0 0
2 2 1 2 0
1 2 0
, ,
, , 0,
, , , 0, 3k k k k k m
y Ay F t y t x
y Ay f t y y t
y Ay f t Y f t Y y t k− −
= + =
= + =
= + − =
(4.3)
( ) ( ) ( )1 0, , , 0, 2m mz Az f t z Y f t Y z t m−= + + − = (4.4)
As in prior section, the underlying matrices in linear recursive equations (4.1) depend upon solutions of the
preceding equations and, in turn, upon 0t and 0 , 2x k , whereas the underlying matrix in (4.3) equals
, 1A k . Hence, for (4.3) the transition matrix is a block-diagonal which expedites repeated computations of
solution to these equations emanated from a set of initial vectors.
Note that for 1m = the corresponding equations are the same in both cases. Also in both sets of equations
( )0 0, ,k ky y t t x= and ( )0 0, ,z z t t x= since , 1ky k and z implicitly depend upon both 0t and 0x as
parameters while complying with zero initial conditions. To shorten the notation we will frequently write below that
( )0,k ky y t x= and ( )0,z z t x= .
5. Behavior of Approximate Solutions on Long Time-Intervals
This section examines behavior of ( )0,ky t x , on long time-intervals playing a key role in our subsequent
analysis. Let us assume firstly that ( )0,ky t x are defined by (4.1) and write the homogeneous counterpart of these
linear equations as follows,
( )( )1, , 1k k k my A f t Y y k−= − (5.1)
Next, we set that k kv = − be the upper general exponents for (5.1) and ( )0, ,k mW t t k are the transition
matrices for these equations. Hence, ( ) ( )1 0 0, ,W t t W t t= , and ( ) ( )0
0, kv t t
k kW t t N e− −
0 , 0kt t N .
5.1. Application of Lipschitz Continuity Condition
Let us assume in this subsection that the following Lipschitz - like continuity conditions hold for 0t t ,
( ) ( ) ( )2 11 2 1 2 2 3, , ,, nf t s f t s l t s s ss− − (5.2)
10
( ) ( ) 43 ,, nf t s l t s s (5.3)
where , 3,4i i = are some bounded neighborhoods of 0x and ( ) 0ˆ , , 2,3k kl t l t t k = .
To simply further notations we set that 1 2 3 4 = , where is a bounded neighborhood of
0x . Next, we acknowledge some essential properties of ( )0,ky t x and exponents kv in the following theorems.
Theorem 1. Assume that 0 0F = ,
0x is sufficiently small value, 1 0v , ( )0, , 1ky t x k are solution to
(4.1), inequalities(1.4), (5.2) and (5.3) hold, and ( ), n nf t x is a continuous matrix in t and x . Then, 1 kv v=
, ( )0
0lim , 0kt t
y t x →
= , ( ) ( )0 0ˆ, ,k ky t x y t x ( )0O x= and ˆ
ky increases in 0x for 0t t .
Proof. We prove this statement by induction. Let 1k = , then, since 0 0F = , ( ) ( )1 0 1 0 0, ,y t x W t t x= and
( ) ( ) ( )( )1 0 1 0 0 1 1 0 0ˆ, , exp ,y t x y t x x N v t t t t = − − . Hence, ( )1 0lim , 0
ty t x
→= ,
( )1 0 0, ,y t x t t for sufficiently small 0x and, due to (5.3), ( )( ) ( )1 0 3 1 0
ˆ, , ,f t y t x l y t x ,
which implies that ( )( )0
1 0lim , , 0t t
f t y t x →
= . Thus, due to Statement 2, 1 2v v= .
Let us next bound ( )2 0,y t x and ( )3 0,y t x to unveil common traits in this sequence. In fact,
( ) 1 0( )
2 0 2,v t t
W t t N e− −
, ( )1 1 1ˆ,f t y l y for sufficiently small 0x , and
( ) ( ) ( )( )0
2 0 2 1 0, , , ,
t
t
y t x W t f t y x d ( ) ( )0
1 2 1 0ˆ , ,
t
t
l W t y x d . Substitution of the prior
bounds in the last integral yields that,
( ) ( ) ( ) ( )1 0
2 0 2 0 21 0 0 0 21ˆ, , , , 0
v t ty t x y t x x e t t t t const
− − = − = , which implies that
( )0
2 0lim , 0t t
y t x →
= .
Next, we infer that, ( ) ( )1 0 2 0 0, , ,y t x y t x t t+ for sufficiently small 0x and, due to (5.3),
( ) ( )( ) ( ) ( )( )1 0 2 0 3 1 0 2 0ˆ, , , , ,f t y t x y t x l y t x y t x + + . The last inequality implies that,
( ) ( )( )0
1 0 2 0lim , , , 0t t
f t y t x y t x →
+ = which with Statement 2 yields that 3 1v v= .
In turn, application of (5.2) and (5.3) infers that ( ) ( )1 2 1 1 2ˆf y y f y l y+ − and
( )1 2 1 2 1 2ˆ ˆf y y l l y y . Then, application of standard integration formulas returns that,
( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 0 1 0 1 022 2
3 0 3 0 31 0 0 0 32 0 33 0ˆ, , ,
v t t v t t v t ty t x y t x x e t t x e t t e t t
− − − − − −= = − + − + ,
3 30 ,i const i = . Hence, ( )0
3 0lim , 0t t
y t x →
= .
Afterward, to simplify the induction let us define functions, ( ) ( ) ( )1 0
0 0 0e ,sv t t
sj jt t p t t t t− −
− = − , where
, 0s j = and ( )0jp t t− is a polynomial in 0t t− of degree j and0p const= which can be zero.
Clearly, ( )0
0lim 0sjt t
t t →
− = .
11
Next, let 1 is a vector space, over which is spanned by ( )0sj t t − and also closed with respect to
application of two operations: multiplication of its entries, i.e.,1 1 2 2 1, , , 1,2s j s j i is j i = and a
convolution, which is defined as follows, ( ) ( )1
0
0 1
tv t
sj
t
e t d
− −− . Note that the last formula follows from
the standard integration rules.
In turn, we assume by induction that,
( ) ( ) ( )0 0 0 0 0
1
ˆ, , , ,kF
i
k k ki K
i
y t x y t x x t t t t k=
= − (5.4)
where kF is k -th Fibonacci number, 1 00 ,ki t t . Clearly, ( )0 1ˆ ,ky t x and increases in
0 0,x t t which implies that ( )0
0lim , 0kt t
y t x →
= .
Next, we show in sequence that (5.4) holds for 1k K= + . In fact, repetition of the foregoing steps yields that,
( )lim , 0Kt
f t Y→
= , which implies that, 1 1Kv v+ = and ( ) ( )1 0
1 0 1,v t t
K KW t t N e− −
+ + . Next, as prior, we infer
that KY for sufficiently small 0x and (5.2) and (5.3) hold under this condition. This implies that,
( ) ( )1 2 2 1ˆ ˆ ˆ, ,K K K Kf t Y f t Y l y l y−− and ( )1 3 1
ˆ, K K K Kf t Y y l y Y− − 3 1 1
ˆˆK Kl y Y − , where
1
1
1
ˆ ˆK
K i
j
Y y−
−
=
= . Consequently, we get that, ( ) ( )1 0 1 0ˆ, ,K Ky t x y t x+ + =
( ) ( ) ( )( )1
0
1 0 2 3 1 0 1 0ˆ ˆ ˆˆ , , ,
tv t
K K K
t
N e y x l l Y x d t t
− −
+ −+ . This implies that ( )0
1 0lim , 0Kt t
y t x+ →
=
and ( )1 0ˆ ,Ky t x+ increases in 0x since integral of positive functions, i.e.,
( ) ( )1
0
0 0
tv t
ki
t
e t d
− −− and
( ) ( ) ( )1
0
0 0 0
tv t
ki sj
t
e t t d
− −− −
Theorem 2. Assume that0 0F , 1 0v and both,
0x and 0F are sufficiently small, ( )0,ky t x is a solution
to (4.1) and inequalities (1.4), (5.2) and (5.3) are intact and ( ), n nf t x is a continuous matrix in both t and x .
Then, 10 , 1k kv v k− , ( ) ( )0
0 0lim ,kt t
y t x O F →
, ( ) ( ) ( )0 0 0 0 0, , ,k ky t x y t x F O x F = + ,
0t t and ( )0 0, ,ky t x F increases in both 0x and 0 0,F t t .
Proof. We proof this theorem by induction. In fact, ( ) ( )( )1 0 0 1 1 0, expy t x x N v t t − −
( ) ( )( ) ( )1 0
1 0 1 0 0 0/ 1 ,v t t
N F v e O x F t t− −
+ − = + and ( ) ( )0
1 0 0lim ,t t
y t x O F →
. Thus, 1y and
( )1 3 1ˆ,f t y l y if both 0x and 0F are sufficiently small. Hence, due to Statement 1, we infer that under
appropriate choice of 0x and 0F , 2 10 v v which, subsequently, yields that,
12
( ) 2 0( )
2 0 2 0, ,v t t
W t t N e t t− −
. Furthermore, since under our assumptions, ( )1 1 1ˆf y l y we get that,
( ) ( )( ) ( ) ( )0 0
2 2 1 0 1 2 1 0ˆ, , , ,
t t
t t
y W t f y x d l W t y x d .
Next, we introduce some notations, which will be used afterward. Let us write the previous formula as follows,
( ) ( ) ( ) ( )2 0 2 0 0 210 0 0 211 0 0, , ,y t x y t x F t t x t t F = − + − , where
( ) ( ) ( ) ( ) ( ) ( )( )1 0 1 02 2
0 0
1
210 0 1 2 211 0 1 2 1, 1
t tv t v tv t v t
t t
t t N N e e d t t N N v e e d
− − − −− − − −−− = − = − . Clearly,
( ) ( )0 0
210 0 211 0lim 0, lim 0t t t t
t t t t const → →
− = − = , and ( )2 0 0, 1, 0,1ij t t i j − = = since they are
defined through integrations of the products of positive functions. Hence, ( ) ( )0
2 0 0lim ,t t
y t x O F →
,
( ) ( )2 0 0 0 0 0, , ,y t x F O x F t t= + and the last function increases in both 0x and 0F , 0t t .
Next, let 2 be a vector space over that is spanned by functions, ( ) ( ) ( )0*
0 0 0e ,s t t
sj jt t p t t t t− −
− = − ,
where 0s , j and ( )0jp t t− is a polynomial in 0t t− of degree j with0p const= which can be zero.
Note that 2 is closed with respect to multiplication of its entries and a convolution which is defined as follows,
( ) ( )0
*
0 ,k
tv t
sj
t
e t d
− −− 0,k mv k .
Next, we assume that,
( ) ( ) ( )0 0 0 0 0 0
1 0
, , , ,kF i
i j j
k k kij K
i j
y t x y t x F t t x F k−
= =
= − (5.5)
where ( ) ( ) ( ) ( )0 0
0 0 2 0 00, , lim 0, , lim 0kij kij kij kiit t t t
t t t t t t i j t t const → →
− − − = − = ,
0t t , kF is k - th Fibonacci number. Clearly, ( )0 0 2, ,ky t x F , ( ) ( )0 0 0 0, ,ky t x F O x F= + ,
0t t , ( ) ( )0
0 0lim ,kt t
y t x O F →
and our assumption holds for . Let us show by induction that (5.5)
holds for 1k K= + . In fact, ( )0, ,k Ky t x k , KY since both0x and 0F are sufficiently small and
( )0 0 0
1 1
,K K
K k K i
k i
Y y Y y O x F t t= =
= = + . This implies that, ( ) 3 2ˆ, k Kf t Y l Y and
( ) ( )0
0lim , Kt t
f t Y O F →
, which, with Statement 1, infers that 10 K Kv v+ .
Next, for sufficiently small 0x and 0F , application of (5.2) and (5.3) implies that,
( ) ( )1 2 1 2ˆ, ,K K Kf t Y f t Y l y− −− and ( )1 3 1 2
ˆ, K K K Kf t Y y l y Y− − . In turn,
( ) ( )
( )( ) ( ) ( )0
1 0 1 0 0
1 1 0 0 2 3 1 0 0
, , ,
ˆ ˆexp , , , ,
K K
t
K K K K
t
y t x y t x F
N v t y x F l l Y x F d
+ +
+ + −
=
− − +
2k =
13
Consequently, ( ) ( ) ( )1
1 0 1 0 0 1, 0 0 0 2
1 0
, , ,KF i
i j j
K K K ij
i j
y t x y t x F t t x F+
−
+ + +
= =
= − , where
( )1, 0K ij t t + − are defined by prior formula, and ( )1 0 0Ky O x F+ = + . Note that ( )1, 0 00,K ij t t t t + −
since multiplication and integration of positive functions yield positive functions and ( )1, 0 2K ij t t + − since 2
is closed with respect to both multiplication of its elements and prior defined convolution. Furthermore, application
of standard integration rules yields that, ( )0
1, 0lim 0,K ijt t
t t i j + →
− = and ( )1, 0lim 0K iit
t t const +→
− = .
Hence, ( )1 0 0, ,Ky t x F+ increases in both 0x and F̂ and ( ) ( )
01 0 0lim ,K
t ty t x O F+
→
Subsequently, we derive the upper bound for the norm of ( )0,ky t x which is determined by formulas (4.3) in the
following,
Theorem 3. Assume that both 0x and 0F are sufficiently small, 1 0v and ( )0, , 1ky t x k are solution to
(4.3) and inequalities (1.4), (5.2) and (5.3) hold. Then,
( ) ( )
( )( )
( )( )( )
0 0 0
1
01 0 01 0 0 1 1 0 0
1 1
, , ,
ˆ / ... exp , 1,1 !
k k
k
k k
k k
y t x y t x F
t t F Fl N x F v v t t k t t
k v v
−
−
=
− − − − − − +
−
(5.6)
where ( )1 2ˆ ˆmax ,l l l= and ( )0 0, ,ky t x F increases in both
0x and0 0,F t t , and series ( )0
1
,k
k
y t x
=
converge absolutely 0t t if 1 0 1/ 1lN F v .
Proof. The first part of this statement can be readily proved by induction. In fact,
( ) ( ) ( ) ( ) ( )( )
( ) ( )( )
1 0 1 0
1 0
1 0 1 0 0 1 0 1 0 1
1 0 0 1 0 1
, , , / 1
/ /
v t t v t t
v t t
y t x y t x F N x e N F v e
N x F v e F v
− − − −
− −
= + − =
= − +
Hence, for sufficiently small 0x and 0F , ( )1 0 0, ,y t x t t , which implies that under these conditions,
( )1 1ˆf y l y .
In turn,
( ) ( ) ( ) ( )
( )( ) ( )( ) ( )
1
0
1 0
2 0 2 0 0 1 1 0 0
2 2 2
1 0 0 1 0 0 1 0 1
ˆ, , , , ,
ˆ / / /
tv t
t
v t t
y t x y t x F N l e y x F d
N l x F v t t F v e F v
− −
− −
= =
− − − +
and, subsequently, under the assumption of this statement, ( )2 0 0, ,y t x t t . In sequence, we assume that
( )0, , 2k Ky t x k are bounded from above by (5.6) and show that this formula holds for 1k K= + .
Clearly, in this case, both Ky and KY for sufficiently small 0x and 0F . In turn,
( ) ( )1, ,K Kf t Y f t Y −− ( )0 0ˆ ˆ , ,K Kl y ly t x F and consequently, ( )1 1 0 0, ,K Ky y t x F+ + =
( ) ( )1
0
1 0 0ˆ , ,
tv t
K
t
N l e y x F d
− −
. Then, application of standard formulas infers the first part of this statement.
14
Next, let us show by induction that ( )0 0, , , 1ky t x F k increases in both 0x and
0 0,F t t . In fact, the
statement holds for 1k = . Assume, subsequently, that ( ) ( ) ( )0 0 0 1 0 2, ,k k ky t x F x t F t = + , 2 Kk
where continuous ( ) 00,ki t t t , 1,2i = . Then,
( ) ( ) ( )
( ) ( ) ( ) ( )
1
0
1 1
0 0
1 0 0 1 0 0
1 0 1 0 1
ˆ, , , ,
ˆ
tv t
K K
t
t tv t v t
K K
t t
y t x F N l e y x F d
N l x e d F e d
− −
+
− − − −
= =
+
Obviously, the integrals of the products of positive functions entering the above formula are positive, which
embraces the second part of this statement.
Finally, to prove absolute convergence of series, ( )0
1
,k
k
y t x
=
, we apply the Cauchy’s root test to (5.6). Clearly,
( ) ( )1
0lim / 1 ! 0k
kt t k
−
→
− − =
and ( )( )1 0 0exp 1,v t t t t− − which simplifies the test and infers our
statement
5.2 Application to Polynomial Systems
In appendix of this section we adopted for simplicity the notation used prior in subsection 5.1 in the matching
statements. Note that functions, ( ) ( )0 0 0ˆ , , , ,k ky t x y t x F and ( )0 0, ,ky t x F play the same role in both
subsections but their definitions in this subsection are different from the prior ones.
Let us assume next that ( ),f t x with ( ),0 0f t = is a vector – polynomial in x without linear components.
Then application of inequalities (1.4), (5.2) and (5.3) to theorems 1 -3 turns out to be redundant which let to void a
previously essential constrain that 0x is sufficiently small. This leads to the following counterparts of theorems 1
- 3.
Theorem 4. Assume that ( ),f t x with ( ),0 0f t = is a vector – polynomial in x without linear components,
0 0F = , 1 0v and ( )0, , 1ky t x k are solution to (4.1). Then, 1 kv v= , ( )0lim , 0kt
y t x→
= ,
( ) ( ) ( )0 0 0 0ˆ, , ,k ky t x y t x O x t t = and ˆ
ky increases in 0x , 0 , 1t t k .
Proof. The proof of this statement is analogous to the previous one with some exceptions. The initial step of
induction and formula (5.4) are intact in this case but kF here is the number of additions in (5.4). To proof the
induction, we need to show that under the assumptions of this theorem, ( )0
lim , 0Kt t
f t Y →
= and
( ) ( )* 1ˆ, ,K Kf t Y f t Y ,where ˆ
K KY Y = ( )0 1
1
ˆ ,K
i
i
y t x=
, ( )*ˆ, Kf t Y is a scalar polynomial in ˆ
KY
and ( )* ,0 0f t = . In fact, these inferences directly follow from repeated applications of standard inequalities that
are briefly discussed in section 2.3, see also Appendix of this paper for the similar derivations.
Firstly, we note that the entries of Jacobian matrix, ( ), Kf t Y , i.e. ( )1, , , ,ij K nf t Y i j are polynomials in KY
of degrees greater or equal to one with ( )1, ,0 0ijf t = . Using mentioned above inequalities we infer that
( ) ( )1 2,, ,ij K ij Kf t Y f t Y , where2,ijf are scalar polynomials in KY of degrees greater or equal to one and
15
( )2, ,0 0ijf t = . In turn, ( ) ( ), ,K K Ff t Y f t Y ( )
1/2
2
2,
, 1
,n
ij k
i j
f t Y=
and, due to assumption of the
induction, ( )0 0
0
1
ˆ ˆlim lim lim , 0K
K K it t t t t
i
Y Y y t x → → →
=
= = , which implies that ( )0
lim , 0Kt t
f t Y →
= .
Next, application of the mentioned above inequalities yields that, ( ) ( ) ( )1
1
, , ,n
K K i K
i
f t Y f t Y f t Y=
=
( ) ( )* *ˆ, ,K Kf t Y f t Y , where *f is a scalar polynomial in ˆ
KY with degree greater or equal two. Hence,
( )* 1ˆ, Kf t Y .
Obviously, ( )* 1 1ˆ, Kf t Y − as well and ( ) ( )1 3 1 1
ˆˆ, ,K K K Kf t Y y y f t Y− − , where 3f is a scalar
polynomial in 1ˆKY −
Theorem 5. Assume that ( ),f t x is a vector – polynomial in x without linear components and ( ),0 0f t = ,
0 0F is a sufficiently small value, 1 0v and ( )0, , 1ky t x k are solution to (4.1). Then,
10 , 1k kv v k− , ( ) ( )0
0 0lim ,kt t
y t x O F →
= , ( ) ( ) ( )0 0 0 0 0 0, , , ,k ky t x y t x F O x F t t = +
and ( )0 0, ,ky t x F increases in both 0x and 0 0, , 1F t t k .
Proof. The proof of this statement is analogous to proofs of theorems 2 and 4. In fact, the initial step of induction
and formula (5.5) are intact but, as prior, in this case kF is the number of terms in (5.5). To embrace the induction,
we need to show that under the assumptions of this theorem, ( ) ( )0
0lim , kt t
f t Y O F →
= and
( ) ( )*
2, ,k kf t Y f t Y , where k kY Y = ( )0 0 2
1
, ,k
i
i
y t x F=
, ( )* , kf t Y is a scalar polynomial
inkY , ( ) ( )*
0 0 0, ,kf t Y O x F t t= + and ( ) ( )*
0lim , kt
f t Y O F→
= . The proofs of these statements are
similar to ones made in theorem 4. In fact, the induction assumes that, ( ) ( )0
0 0 0lim , ,kt t
y t x F O F →
= ,
( ) ( )0 0 0 0 0, , , ,k Ky t x F O x F t t k= + and ( )0 0, ,ky t x F increases in0x and 0F
0 , Kt t k . Thus, application of the steps used to prove the previous statements infers this theorem
Theorem 6. Assume that ( ),f t x with ( ),0 0f t = is a vector – polynomial in x , 1 0v and ( )0,ky t x ,
1k are solution to (4.3). In addition, we assume that: 1. 0 0F = or 2. 0 0F . Then, 1. ( )0lim , 0kt
y t x→
= ,
( ) ( ) ( )0 0 0, ,k ky t x y t x O x = and ky increases in 0x for 0t t . 2. ( ) ( )0 0lim ,k
ty t x O F
→ ,
( ) ( ) ( )0 0 0 0 0, , ,ky t x y t x F O x F = + and ( )0 0, ,y t x F increases in both 0x and 0 0,F t t .
Proof. In fact, in this case the ‘linear block’ of (4.3) is defined by matrix ( )A t which immediately yields that
1, 1kv v k= . Thus, we merely need to show that: 1. ( ) ( ) 1ˆ, ,K Kf t Y f t Y+ if 0 0F = and 2.
( ) ( ) 2ˆ, ,K Kf t Y f t Y+ if 0 0F , where both f+ and f +
are scalar polynomials in ˆKY . The proofs of
16
these statements are directly followed from mentioned above inequalities and are completely analogous to the prior
ones
Clearly, under conditions of theorems 1-6, ( ) ( )0 0
0 0lim , lim ,t t t t
x t x z t x → →
= if0 0F = and
( ) ( ) ( )0 0
0 0lim , limt t t t
x t x z t O F → →
= + if 0 0F . Hence, behavior of the norm of error term, ( )z t on long
time intervals is circumstantial in analysis of boundedness and stability of solutions to (1) and (2), respectively.
6. Linear Error Equation and Its Applications
This section applies the methodology developed in [43] to estimate the error norm of approximate solutions given
by (4.1), i.e., the norm of solution to (4.2) using Lipschitz continuity condition. This, in turn, yields successive
approximations to bilateral solution bounds and boundedness/stability conditions for solutions of equations (1) and
(2). Our inferences can be readily attuned if the error term is defined by equation (4.4) as well.
6.1. Linear Error Equation
Firstly, we apply inequality (2.8) to equation (4.2) as follows,
( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 0 1 0, , , , , 0m m m mD z p t z c f t z Y f t Y f t Y y t t z tt+
− − + + − − = (6.1)
where functions, ( )p t and ( ) ˆc t c are defined in section 2, see also [43]. Clearly, 1z z and both,
mY and 1 0,mY t t− if 0x and 0F are sufficiently small. We will show below that this condition also
yields that, 1 0,mz Y t t+ as well. Hence, application of standard norm’s inequality and (5.2) transforms (6.1)
into the following form,
( ) ( )
( )
1 1 0 0
1 0 0
, ,
, 0
D z t z t x t t
z t x
+ +
=
where
( ) ( ) ( ) ( ) ( ) ( )( )0 2 0 1 0, , , ,m m mt x c t l t y t x f t Y y t x− = + (6.2)
and ( ) ( ) ( ) ( )2t p t c lt t = + . Subsequent application of comparison principle to the last inequality leads to the
following linear equation,
(6.3) ( )
( )
1 1 0
1 0 0
,
, 0
Z t Z t t
Z t x
= +
=
where, ( ) ( )1 0 1 0 0, , ,Z t x z t x t t . Let us also recall that both ( )1 0,z t x and ( )1 0,Z t x depend implicitly upon
both 0t and 0x but assume zero initial condition at 0t t= .
Due to our initial assumptions, ( ) ( ),p t c t , ( )2l t and, hence, ( )t are continuous and bounded functions in t .
In addition, under assumptions of either Theorems 1 ( 0 0F = ) or Theorem 2 ( 0 0F ), we infer that continuous
0, t t . In the latter case we infer that,
( ) ( ) ( )( )0 0 0 0 2 3 1 0 0ˆ ˆˆ, , , , , ,m mt x F cy t x F l l Y t x F− = + (6.4)
and under assumptions of Theorem 2, we get that, ( )0 0 2, ,t x F , ( ) ( )0
0 0 0lim , ,t t
t x F O F →
= ,
( )0 0, ,t x F increases in both0x and 0F , 0t t and, ( ) ( )
0
0 0 0 0sup , ,t t
t x F O x F
= + since
( ) ( )0
0 0 0 0sup , ,t t
y t x F O x F
= + .
17
In turn, for 0 0F = (6.4) should be attuned as follows,
( ) ( ) ( )( )0 0 0 2 3 1 0 0ˆ ˆ ˆ ˆˆ ˆ, , , , ,m mt F x ky t x l l Y t x F− = + (6.5)
and under the assumptions of Theorem 1, ( )0 1ˆ ,t x , ( )
00
ˆlim , 0t t
t x →
= , ( ) ( )0
0 0ˆsup ,
t t
t x O x
= and
( )0ˆ ,t x increases in
0x , 0t t .
Subsequently, the solution to (6.3) can be written as,
( ) ( ) ( )0
1 0 0 0 0, exp , , ,
t t
t
Z t x s t ds x d t t
=
(6.6)
6.2. Application of Linear Error Equation
Thereafter, we present the following theorems which provide bilateral bounds and corresponding
boundedness/stability criteria for solutions to (1) or (2), respectively.
Theorem 7. Assume that booth, 0x , 0 0F are sufficiently small, ( ) ˆ,t − 0
ˆ, 0t t , ( ),f t x
is a continuous matric, inequalities (1.4), (5.2) and (5.3) hold and ( ) ˆc t c . Then, ( )0 0, ,x t x t t ,
( ) ( )0
0 0lim ,t t
x t x O F →
= , where ( )0,x t x is a solution to (1) and
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )0 1 0 0 0 1 0 0 0 1 0
0 0 1 0 0 0
, , , , , , , ,
, , , , ,
m m m
m
Y t x Z t x F Y t x Z t x x t x Y t x Z t x
Y t x F Z t x F t t
− − +
+ (6.7)
where ( )1 0 0, ,Z t x F = ( ) ( )
0
ˆ
0 0, ,t
t
t
e x F d
− −
and both left – side inequalities should be adjusted to zero if
they assume negative values.
Proof. In fact, at the end of section 2, we indicated that, 1
ˆ 0v . Then, due to Theorem 2,
( ) ( ) ( )0 0 0 0 0 0
1
, , , ,m
m m k
k
Y t x Y y t x F O x F t t=
= = + and ( ) ( )0
0 0lim ,mt t
Y t x O F →
, which
implies that under conditions of this statement mY .
Next, to embrace the application of (5.2) to (6.1), we will show by contradiction that, 0,mz Y t t+ if 0x
and 0F are sufficiently small. In fact, since ( )0 0z = , let us assume that, 1 0t t such that, ( ) ( ) ( )1 1z t y t+ ,
where ( ) is a boundary of . Then, due to (6.6),
( ) ( ) ( ) ( ) ( )1 1
0 10 0
ˆ ˆ
1 0 0 0 0 0 0 1, , , sup , [ , ]
t t
t t
t t tt t
Z t x e x F d e d O x F t t t
− − − −
= + .
This implies that ( )1 0,Z t x can be made arbitrary small 0 1[ , ]t t t if 0x and 0F are sufficiently small which
infers that, ( ) ( )1 1z t y t+ − under the assumptions of this statement.
In addition, (6.6) implies that,
( ) ( ) ( ) ( ) ( ) ( )
0 0
ˆ ˆ
1 0 1 0 0 0 0 0 0 2 0, , , , , ,t t
t t
t t
Z t x Z t x F e x F d O x F e d t t
− − − −
= = + ; hence,
( )1 0 0, ,Z t x F increases in both 0x and 0F for 0t t . In turn, ( ) ( )0 1 0, ,z t x z t x ( )1 0,Z t x
18
0, t t and ( ) ( )0
0 0lim ,t t
z t x O F →
= . Consequently, ( )0,x t x , ( ) ( )0
0 0lim ,t t
x t x O F →
= and
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )0 1 0 0 0 1 0 0 0 1 0
0 0 1 0 0 0
, , , , , , , ,
, , , , ,
m m m
m
Y t x Z t x F Y t x Z t x x t x Y t x Z t x
Y t x F Z t x F t t
− − +
+
Theorem 8. Assume that 0 0F = , 0x is sufficiently small, ( ),f t x is a continuous matric,
( ) 0ˆ ˆ, , 0t t t − , inequalities (4.1), (5.2) and (5.3) hold and ( ) ˆc t c . Then, the trivial solution to
(2) is asymptotically stable and
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
0 1 0 0 0 0 0 1 0
0 1 0 0
ˆ, , , , , , ,
ˆ ˆ, , ,
m m m
m
Y t x Z t x Y t x Z t x x t x Y t x Z t x
Y t x Z t x t t
− − +
+
(6.8)
where ( ) ( )0 0
1
ˆ ˆ, ,m
m i
i
Y t x y t x=
= , ( ) ( ) ( )0
ˆ
1 0 0 ˆ ˆ, ,t
t
t
Z t x e x d
− −
= and both left – side inequalities
should be adjusted to zero if they assume negative values.
Proof. In fact, in this case, due to Theorem 1, ( ) ( )0 0 0, ,mY t x O x t t and ( )0
0lim , 0mt t
Y t x →
= .
As in the prior statement, we can readily show that both1
1
1
m
i
i
y−
=
and1
m
i
i
z y=
+ , 0t t for sufficiently small
0x . Hence, ( ) ( ) ( ) ( )0
0
ˆ
1 0 1 0 0 1ˆ ˆ, , ,
tt
t
Z t x Z t x e x d
− −
= since, ( )0 1ˆ ,t x . This implies that,
( )0
0lim , 0t t
z t x →
= and that (6.8) holds
Note that in (6.7) and (6.8) most conservative upper bounds depend only upon scalar parameters, 0x and 0F or
0x , respectively, whereas in most conservative lower bounds just the estimate of error function depend upon
these scalar parameters.
Let us assume, in turn, that the approximate solution ( )0,mY t x is defined by (4.3). Clearly, in this case equation
(6.3) holds with the same ( )0,t t and formulas (6.2), (6.4) and (6.5) hold as well with obvious simplifications.
Thus, theorems 7 and 8 hold in this case as well.
7. Nonlinear Error Equation and Its Applications
Utility of linear error equation (6.3) enables derivation of local stability and boundedness criteria but fails to
estimate trapping/stability regions of the original equations. This section develops a nonlinear version of the error
equation via application of a nonlinear extension of Lipschitz inequality, see [43] and section (2.1) of this paper for a
brief introduction of this concept. Such nonlinear error equation frequently better reflects a nonlinear character of
the underlying equations and enables estimation of the trapping/stability regions for systems (1) or (2), respectively.
7.1. Nonlinear Error Equation
To streamline further derivations, we assume in this section that ( ),f t x is a vector polynomial in x without
linear term and ( ),0 0f t = . Under these conditions, (4.2) can be written as follows,
19
( ) ( )
( )0 0
, ,
, 0
z A t z t y z
z t x
= +
= (7.1)
where 1,...,T
my y y= , ( ), ,t y z is a polynomial in z with coefficients depending upon t and y . Note that
in the reminder of this section we assume that ( )0,y t x is defined by either (4.1) or (4.3) under conditions of
Theorem 4/Theorem 5/Theorem 6, respectively. Thus, ( ) ( ) ( )0 0 01
ˆ, , ,my t x y t x Y t x =
( )0 1
1
ˆ ,m
i
i
y t x=
if 0 0F = and ( ) ( ) ( )0 0 0 0 0 2
1
, , , , ,m
m i
i
y t x Y t x F y t x F=
= if 0 0F . In turn,
successive application of standard norm inequalities (see section 2.3 and Appendix of this paper) yields that,
( ) ( ) ( )* *, , , , ,t y z t y z t y + (7.2)
where ( ) ( )* 0, , , ,t y z t x z = is a scalar-polynomial in z , ( )0, ,0 0t x = , ( ) ( )*, ,0 ,t y t y =
( )0,t x= and ( )* ,t y is a scalar polynomial in y with ( )* ,0 0t = . Additionally, ( ) ( )* *ˆˆ, , mt y t Y
( )0 1ˆ ,t x= if 0 0F = and ( ) ( ) ( )* * 0 0 2, , , ,mt y t Y t x F = if 0 0F . Thus,
( )0
0lim , 0t t
t x →
= if 0 0F = and ( ) ( )0
0 0lim ,t t
t x O F →
= if 0 0F , see Appendix for additional details on
definitions of *̂ and * .
Consequently, application of (2.12) to (7.1) with account of (7.2) returns the following inequality,
( ) ( ) ( ) ( ) ( )
( )
2 2 0 2 0 0
2 0 0
, , , ,
, 0
D z p t z c t t x z c t t x t t
z t x
+ + +
= (7.3)
Clearly, (7.3) implies that 2z z .
Subsequent applications of the comparison principle to (7.3) yields the following scalar equation,
( ) ( ) ( ) ( ) ( )
( )
2 2 0 2 0 0
2 0 0
, , , ,
, 0
Z p t Z c t t x Z c t t x t t
Z t x
= + +
= (7.4)
where2 2z z Z . In the reminder of this paper we assume that (7.4) admit a unique solution for 0t t .
7.2. Estimation of Bilateral Solutions’ Norms and Trapping/Stability Regions
The following statements infer some properties of solutions to multidimensional equations (1) and (2) under
assumptions set on solutions to scalar equations (7.4). We assume in these statements without repetition that
( )0
nt is a bounded neighborhood of 0x .
Theorem 9. Assume that ( ),f t x is a vector polynomial in x without linear component, ( ),0 0f t = , 0 0F is
a sufficiently small value, 1 0v , ( )0, ,k my t x k are solutions to (4.1) , ( ) ( )0
2 0 0lim ,t t
Z t x O F →
=
( )0 0, x t and ( )0,x t x is a solution to (1).
Then, ( )0 0, ,x t x t t , ( ) ( ) ( )0
0 0 0 0lim , ,t t
x t x O F x t →
= and is included in the trapping
region of (1). Furthermore, under the above conditions, solutions to (1) assume the following bilateral bounds,
( ) ( ) ( ) ( ) ( )0 2 0 0 0 2 0 0, , , , , ,m mY t x Z t x x t x Y t x Z t x t t− + (7.5)
where the left side of (7.5) should be adjusted to zero if it takes negative values.
20
Proof. Really, in this case due to Theorem 5, 2 0,m mY Y t t and, in turn, ( )
00lim ,m
t tY t x
→
( ) ( )0
0 0 0lim , ,mt t
Y t x F O F →
= . Hence, under conditions of this statement, (7.5) follows from (1.10) which,
subsequently, implies that ( ) ( )0 0 0 0, , ,x t x x t t t and ( ) ( )0
0 0lim ,t t
x t x O F →
= , ( )0 0x t
Theorem 10. Assume that ( ),f t x is a vector polynomial in x without linear component, ( ),0 0f t = , 0F = ,
1 0v , ( )0, ,k my t x k are solutions to (4.1), ( )0,x t x is a solution to (2) and ( )0
2 0lim , 0t t
Z t x →
= ,
( )0 0x t .
Then, the trivial solution to (1.2) is asymptotically stable, ( )0t is included in the stability region of the trivial
solution to (1.2) and ( )0,x t x admits bilateral bounds (7.5) with 0F = .
Proof. In fact, in this case, due to Theorem 4, ( ) ( )0
0 0 0lim , 0,mt t
Y t x x t →
= , hence, the above theorem
follows from (1.10) and inequalities (7.5) with 0 0F =
Theorem 11. Assume that ( ),f t x is a vector polynomial in x without linear component, ( ),0 0f t = , 1 0v ,
( )0, ,k my t x k are solutions to (4.3). Assume also that: 1. ( ) ( )0
2 0 0 0lim , 0,t t
Z t x x t →
= if 0 0F =
and in this case ( )0,x t x is a solution to (2); 2. ( ) ( )2 0 0lim ,t
Z t x O F→
= ( )0 0, x t if 0 0F and in this
case ( )0,x t x is a solution to (1).
Then, 1. the trivial solution to (1.2) is asymptotically stable, ( )0t is included in the stability region of the trivial
solution to (1.2) and ( )0,x t x admits bilateral bounds (7.5) with 0F = . 2. ( )0, ,x t x 0t t ,
( ) ( )0
0 0lim ,t t
x t x O F →
= and ( )0t is included in the trapping region of (1) containing 0x and ( )0,x t x
admits bilateral bounds (7.5).
Proof. The proof of this statement is based on Theorem 6 and is analogous to the proofs of the previous two
statements.
7.3 Approximate Solution to Error Equation
Though (7.4) is non-integrable scalar equation, the second and third additions in the right-hand side of this
equation are nonnegative, whereas the first - can be either positive/negative or change sine for some values of t . To
exploit this property, we set that ( ) ( ) ( )0 2 0 2 0 2, , , , ,t x Z D t x Z t x Z− = + , where ( )0 00 , ,D t x t t ,
( )0 2, ,t x Z− is a scalar polynomial in 2Z without linear component and with ( )0, ,0 0t x− = . In turn, setting
( )0 2, , 0t x Z− = yields a linear, nonhomogeneous and scalar equation,
( ) ( ) ( )
( )
3 1 0 3 0 0
3 0
, , ,
0
Z P t x Z c t t x t t
Z t
= +
= (7.6)
where ( ) ( ) ( ) ( )1 0 0, ,P t x p t c t D t x= + . Clearly, this equation possesses a unique solution
( ) ( ) ( ) ( )0
3 0 1 0 0, , , ,
t
t
Z t x t x c x d = (7.7)
21
where ( ) ( )1 0 1 0, , exp ,
t
t x P s x ds
= . Apparently, ( ) ( )3 0 2 0 0, , ,Z t x Z t x t t , and ( )2 0,Z t x
( )3 0,Z t x should be close to each other if ( )0 2, ,t x Z− is sufficiently small. Clearly, if ( )0
3 0lim ,t t
Z t x →
= ,
( )0 0x t , then ( ) ( )0
2 0 0 0lim , ,t t
Z t x x t →
= as well. Thus, ( ) ( )0 0t t , where is an
estimate of the trapping/stability region attained through evaluation of (7.4). Note also that ( )0t is enclosed in
actual the trapping/stability region, whereas ( )0t - ought not be included in such region. Yet, our simulations
show that ( )0t provides a valuable assessment of the actual trapping/stability regions which can be readily
obtained by utility of (7.7).
Finally, we present a sufficient condition under which ( )0
3 0lim , 0t t
Z t x →
=
Theorem 12. Assume that ( ) ( ) ( ) ( )0
0
1
0 1 0 1 0 0 1 0 0lim sup , , 0, 0,
t
n
t tt
t t P s x ds t x x t −
→− = − ,
( ) 0,c t c t t and ( )0, ,k my t x k are defined by either (4.1) or (4.3). Then, 1.
( ) ( )0
3 0 0 0lim , 0,t t
Z t x x t →
= if 0 0F = and 2. ( ) ( ) ( )0
3 0 0 0 0lim , ,t t
Z t x O F x t →
= if 0 0F .
Proof. In fact, ( )1 0x defines the Lyapunov exponent for the homogeneous part of (7.6). Hence,
( ) ( )( )( )1 0 1 1, , expt x D t = + − , where constants 1D and 0 , see section 2.1 of this paper and
further citation therein. In turn, ( ) ( )( )( ) ( )0
3 0 1 1 0, exp ,
t
t
Z t x D c t x d + − . Thus,
( )( )( ) ( ) ( )( )( ) ( )0 0
1 0 1 0 1 0ˆexp , exp , ,
t t
t t
t x d t x d t t + − + − if 0 0F = and
( )( )( ) ( ) ( )( )( ) ( )0 0
1 0 1 0 0 2 0exp , exp , , ,
t t
t t
t x d t x F d t t + − + − if 0 0F
8. Simulations This section applies the developed above methodology to estimate the bounds of solutions and the
trapping/stability regions for the Van der-Pol and Duffing -like models featuring typical dissipative and conservative
nonlinearities.
8.1 Van der Pol – like Model
Firstly, we simulate system (3.1) with the following values of parameters which includes a large nonlinear
coefficient, 1 2 1 2 1 22, 1.2, 100, 5, 4.8 , 21a a r r = = = − = = = = . Note that a equals either zero or
0.23 in this set of simulations. For system (3.1) with 0a = and a stable trivial solution typical time-histories of
upper estimates of the norms of error terms are plotted in Figure 1 for initial vectors locating either within or beyond
its stability region.
22
Fig. 1a Fig. 1b
Fig. 1 Time histories of the estimates of the norm of
error term computed for initial values within (Figure 1a)
and beyond (Figure 1b) stability region of homogeneous
counterpart of (3.1)
Fig. 2a Estimation of stability region
of the trivial solution of (3.1) by the
first approach
Fig. 2b Estimation of trapping
region for (3.1) by the first approach
Fig. 3a Estimation of a larger
size stability regions for system
(3.1) by first approach
Fig. 3b Estimation of a larger
size stability regions for (3.1)
by second approach
Fig. 2c Estimation of stability region of the trivial solution of (3.1) by second approach
Fig. 2d. Estimation of trapping region for (3.1) delivered by second approach.
23
In both cases the estimates are stemmed from zero and approach either zero (Figure 1a) or infinity (Figure 1b) with
t → . If 0a and 0x is fitted into the trapping region of (3.1), the estimates approach ( )O a ; if in this case
0x is located beyond the trapping region, then the estimate is approach infinity in our simulations. The threshold
sets of 0x differentiating these patterns of behavior of solutions to a scalar equation (7.4) yield the estimate of the
boundaries of trapping/stability regions of (3.1) in our simulations.
Fig. 2 compare three estimates to the boundaries of the corresponding regions which are developed consecutively
by simulating equations (3.2) and (3.4) (blue – lines), (3.5) and (3.7) (green -lines) and (3.8) and (3.10) (red - lines).
As is mentioned in section 3, the complete set of equations in the last two cases include also the appropriate
equations derived at the previous steps. Black – lines mark the reference estimates of actual boundaries of the
corresponding regions, which are obtained in simulations of (3.1). Aqua – lines mark approximations delivered by
linearized equation (3.10). These lines are located beyond the regions encircled by red – lines. Figure 2a and 2b display reference and approximate boundaries of trapping/stability regions for (3.1)
corresponding to our first approach. Figures 2c and 2d display in matching colors the corresponding approximations
delivered by the second approach that is defined by equations (3.2), (3.11), (3.12) and (3.14). Note that in Figures 2a
and 2c, 0a = and in Figures 2b and 2d, 0.23a = . Clearly, successive approximations considerably enhance the
accuracy of the corresponding estimates on every consecutive step for both approaches. More efficient in
computations second approach provides less accurate estimates than its more elaborate counterpart for every used
iteration. In the first approach, solutions to linearized and integrable error equations (aqua – lines) provides the
estimates that are sufficiently close to the reference ones, but the differences between these estimates increase for
less accurate second approach. Note that in all these cases the linearized equations provide excessive estimates of
the trapping/stability regions.
Figure 3 shows application of the first approach to estimation of a larger size stability region of equation (3.1). In
these simulations, 1 21 1, = = − , 0a = and other values of parameters are equal to the previous ones. Literally,
the accuracy of our approximations remains intact in these cases as well.
Fig. 4a displays simulated time –
histories of the accurate and
approximate solutions to homogeneous
counterpart of (3.1) as well as the
running low and upper bounds of the
accurate solution stemmed from initial
vectors located within stability region of
(3.1) near its boundary
Fig. 4b displays the simulated time – histories of the accurate and
approximate solutions and low and
upper bounds of the accurate solution stemmed from the initial vector located
in the central part of stability region of
homogeneous counterpart of (3.1)
Fig. 4c displays simulated time – histories
of the accurate and approximate solutions
and running upper and lower bounds of the
accurate solution stemmed from initial
vector located within the trapping region of
(3.1).
24
Simulated time -histories of the norms of accurate and approximate solutions to (3.1) as well as their lower and
upper bounds are displayed in Figure 4. Figures 4a and 4b show the results of simulations of (3.1) with 0a =
whereas Figure 4c display simulations made with 0.23a = . In all these three cases the difference between the
norms of approximate and authentic solutions remain small if the initial vectors are located within the major parts of
the trapping/stability regions but near its boundary. Consecutively, black and red – lines, which display simulated
time – histories of the norms of accurate and approximate solutions, are practically overlaid on these figures. The
running upper and lower bounds of the estimates of norms of accurate solutions are plotted in dashed green and blue
lines.
Note that the lower bounds should be adjusted to zero if they take negative values. In fact, these estimates turn out
to be less accurate if the initial vector is located near the region’s boundary, see Figure 4a, but the accuracy provided
by these bounds increases substantially if the initial vector is moved into a central part of stability region, see Figure
4b. Figure 4c plot the corresponding bounds and the norms of accurate and approximate solutions to (3.1) stemmed
from the initial vector locating within the trapping region of the “forced solution” to this equation. In all of these
three cases time -histories of the estimates of the error – norms relatively quickly reach their maximal value and
subsequently approach either zero or ( )O a if t → .
Analysis of behavior of the norms of approximate solutions in narrow neighborhoods of the boundaries of
trapping/ stability regions spotlight some distinctiveness that can be used to estimate the boundaries of these regions
via bypassing utility of (7.4). In fact, Figures 5a and 5b display behavior of the norms of accurate and approximated
solutions emanated from initial vector locating within and beyond the stability region of homogeneous counterpart
of (3.1) near its boundary. In both cases the norms of solutions are practically overlaid on initial time intervals but
diverge thereafter. Figure 5c demonstrates that the maximal value of the norm of an approximate solution to this
equation rapidly increases if the corresponding initial vector is moved beyond stability region. Finally, Figure 5d
plots in red, black and blue – lines time -histories of the norms of approximate solutions stemmed from initial
vectors located near stability boundary. The initial vector with smallest norm (red – line) is located within the
stability region of this equation near its boundary, whereas two other initial vectors are located beyond the boundary
of this region. In turn, first two local maxima are practically equal on the black – line, whereas on blue line, second
maxima exceed the first one. Hence, an estimation of the boundary of the trapping/stability region can be rendered
by selecting such initial vectors which practically equate first two local maximal values of time-histories of the
corresponding norms of approximate solutions. Application of this heuristic criterion let us to skip simulation of
Fig. 5a displays in blue and aqua
- lines simulated time – histories
of the accurate and approximated
solutions emanated from initial
vectors locating within the
stability region of homogeneous
counterpart of (3.1) near its
boundary
Fig. 5b displays in blue and aqua - lines simulated time – histories of
the accurate and approximated solutions emanated from initial
vector locating beyond the stability
region of homogeneous counterpart of (3.1) near its boundary
Fig. 5c displays simulated
time – history of
approximate solution
emanated from initial
vector locating beyond the
stability region of
homogeneous counterpart
of (3.1)
Fig. 5d displays simulated time
– histories of approximate
solutions emanated from initial
vectors locating within (red –
line) and beyond (black and blue
– lines) stability region of
homogeneous counterpart of
(3.1) near its
boundary
25
(7.4) and in our simulations provides upper estimates of the norms of threshold initial vectors which exceed the
norms of directly simulated ones by about 4% /5%.
8. 2. Duffing – like Model
Fig. 6a plots the estimated
boundaries of stability region for
Duffing system
Fig. 6b plots the estimated
boundaries of trapping region for
Duffing system,0
3.5F =
Fig. 6c plots the estimated
boundaries of stability region
for Duffing system with small
damping
Fig. 6d plots the estimated
boundaries of stability region
for Duffing system, 0
/ 2t =
Fig. 6e plots the estimated
boundaries of stability region for
Duffing system, 0
4t =
Fig. 6f plots the estimated
boundaries of trapping region for
Duffing system, 0
1.5F =
Fig. 6g plots the estimated
boundaries of trapping region
for Duffing system affected by
initial pulse- function
Fig. 6h plots the estimated
boundaries of trapping region
for Duffing system affected by
initial pulse- function
This section applies the developed methodology for estimating the trapping/stability region of Duffing – like
model which was widely used in various applications, see e.g. [47]. The governing equations for this model can be
obtained by replacing the nonlinear term in the second equation of (3.1) as follows,3 3
2 2 2 1x x → . Yet, the
mechanisms involved in developing the trapping/stability regions in Van der-Pol and Duffing – like models are
substantially different, but our methodology works in this case alike. Figure 6 display estimations of
trapping/stability regions for Duffing - like models developed by our first approach which follows the protocol used
prior in section 8.1. Figure 6a displays the estimates of stability boundaries for our model delivered by the first three
approximations for the following values of parameters, 1 2 1 21, 10, 0, 5a a a = = − = = = . Figure 6b displays
the estimates of the trapping region of the forced solution of our model for
26
1 2 1 21, 10, 3.5, 5a a a = = − = = = . Figure 6c shows that the accuracy of our approximations to stability
boundaries decreases for small values of 1 0.05 = . Such behavior is expected since for this system the successive
approximations diverge if 1 0 = . Nonetheless, utility of both extra additions in (1.10) and the transformations
escalating the speed of convergence of these series, increase the accuracy of our approximations in these cases as
well [48].
Figures 6d -6h show the impact of the change in 0t on the shape of stability/trapping regions in our system.
Figures 6a, 6d and 6e are computed for the same set of parameters but in the former case 0 0t = whereas in latter
two cases 0 / 2t = and 4 , respectively. It turns out that the differences in the shapes of the corresponding
stability regions are obvious but rather moderate and consistent with almost – periodic character of variable
coefficients entering this model.
Figures 6f -6h display the impact of the change in 0t on the shape of trapping regions of the forced solution of
Duffing system. Note that in these figures the most left points on the matching curves are labeled for the sake of
comparison and 1 2 0a a= = . In all these cases 0 1.5F = and in Figure 6f, 2 4 = , whereas in Figures 6g and 6h,
( ) ( )2
14t t = + , where ( ) ( )( )1 1 / 2H tt = − − , ( )H t is a Heaviside function and equals either -2 or 4 in
Figures 6g and 6h, respectively. Apparently, the pulse does not affect behavior of this system’s solutions if
0 / 2t , see Figure 6f.
Clearly, in these cases the effect of alteration of 0t on the shapes of trapping regions of the corresponding system
is noticeable but rather moderate as well. Meanwhile, piece-wise constant and other types of time – varying
coefficients can seemingly change the behavior of solutions on some finite/infinite time – intervals which, in turn,
will affect the shapes of trapping/stability regions.
Note also that our estimates tend to smooth the actual boundaries of trapping/stability regions which might have
highly irregular structures [21].
9. Conclusion and Future Work
This paper develops a novel methodology for successive approximation of solutions which are stemmed from the
trapping/stability regions of some nonlinear and time – varying systems. Next, we estimate the norm of error -
component developed by such approximations. This let us to infer enhanced stability/ boundedness criteria, develop
bilateral bounds of the norms of solutions and estimate the trapping/stability regions for such systems.
Two techniques were devised for successive approximations of solutions to original systems: first - conveys more
accurate approximations and second - is more efficient in computations. Subsequently, we provide two approaches
to estimation of the norms of error - components which engage the methodology that we developed in [43]. The first
approach exploits application of the Lipschitz continuity condition which leads to scalar, linear and integrable error
equation. The second approach adopts a nonlinear extension of Lipschitz condition and develops a nonlinear scalar
error -equation with variable coefficients that better reflects the character of the underlined systems and allows
estimation of their trapping/stability regions. Both approaches provide bilateral bounds for the norms of solutions
and boundedness/stability criteria.
We show in inclusive simulations that just a few iterations provided by developed techniques return the
approximations which resemble with high fidelity the time- histories of the norms of accurate solutions stemming
from the trapping/stability regions of original systems. The accuracy of these approximations remains intact for
nonlinearities of different kinds and magnitudes but can be affected if the largest Lyapunov exponent of the linear
block of the corresponding system turn out to be close to zero. Consequently, such approximations enable recursive
estimation of the trapping/stability regions.
Subsequently, we intend to develop upper and lower bounds to solutions of the nonlinear error – equation in close
form, extend our methodology to systems under uncertainty and mitigate some of the technical constrains of the
developed approach. This should allow us to extend this methodology to a wider class of systems arising in various
application domains.
27
Appendix
Let us sketch, for instance, how to derive inequalities (7.2). Assume that, ( ) 1, [ ,..., ]T
nf t x f f= , , , nx u z
, ( ) ( ) ( )( ), ,iidd
i i i i i i if t x t x t z u d = = + . Thus, ( ) ii dd
i i i i i i if z u z u = + +
( ) ( )i id d
i i i i iz u z u + + ( ) id
i z u + .Then, taking into account that,1
f f =1
n
i
i
f=
,
we infer that, ( )*
1
, i
nd
i i
i
t u u =
= , ( )* ,t y = ( ) ( )*
1
,i
nd
i i
i
z u t u =
+ − , ( )1
ˆ , i
nd
i
i
t u u =
= if
0 0F = . Subsequently, we enter in this formula that 1
u u . The prior formula holds if 0 0F with obvious
change in notation.
Assume next that, ( ) ( ), d
id idf t x t x= , where 1
1
1
... , ,n
nddd
n i i
i
x x x d d d=
= = . Then,
( ) 1
1, ... ndd
id id nf t x x x= which enables utility of the prior procedure for every id
ix . Clearly, such sequence
of steps yields (7.2) if ( ),f t x is a vector - polynomial in x .
Acknowledgement. The programs used to simulate the models in section 8 of this paper were developed by Steve
Koblik who also was engaged in discussions of the corresponding results.
Conflict of Interest: The author declares that he has no conflict of interest
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