nonlinear-01-29-14

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Nonlinear Dynamic

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Dynamic systems

• Differential equations:Describe the evolution of the system in continuous time.

(1)

• Iterated maps: Describe the system in discrete time

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Linear and nonlinear systems

• The system (1) is linear, if all xi (x1….xn) appear to the first power only. Otherwise the system is nonlinear (products, powers, & Cos(x)..)

• Linear Easy to solve analytically, (system can be broken into parts).

• Nonlinearity difficult to solve analytically.

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Flows on the line

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1D order system,

• For this system: : is the position of an imaginary particle

moving along the real line. : is the velocity of the particle.

System vector field on the line, and to draw this vector we need to plot vs (Phase portrait).

x

x

x x

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The phase portrait

• The phase portrait of a dynamical system contains a lot of information about the solutions of the system, without all the detail one finds in a solution formula.

- Draw the phase space.- Find the equilibrium points. - Draw them in the phase space.- Determine the motion in between any two

equilibrium points.

0x

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Motion between Fixed Points F.P

• Imagine that a fluid is flowing steadily along x-axis with velocity equal to f(x) then:

- The flow is to right when > 0.- The flow is to left when < 0.- There is no flow when = 0. (F.Ps)• Then the system looking to flow (along

trajectory) to a stable position from right or left.

x

xx

8

yy sin

Draw the phase space:

Find the equilibrium points: 0sin y,.....2,,0 y

Draw them in the phase space

Motion in between F.P

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Trajectory and phase space

IC

(Position, Velocity)Trajectory

Phase space

Our goal is to solve the system by drawing the trajectories without actually solving the system

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Stability Definitions• A fixed point is called stable if solutions that

start near it stay near it. (flow from left & right toward F.P)

• A fixed point is called unstable if solutions that start near it, end up wandering away from it.

(flow from left & right away from F.P)

• A fixed point is called asymptotically stable if solutions that start near it, approach the fixed point as t ---> ∞.

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Flow , 0x

Flow , 0x

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Stability of equilibrium solutions

F.Ps

StableAttractors or Sinks

Flow toward themSolid black dots

UnstableRepellers or Sources

Flow outward themOpen circles

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- Find all fixed Points (F.Ps), by letting f(x)=0.

- Plot f(x), and show all F.Ps

- Specify the flow direction.

- Classify the stability for each F.P

Procedures to analyze stability of F.Ps

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Linear stability analysis

• It’s a way to classify the stability of the F.Psby introducing a small perturbation η(t) away from F.P and check if it grows or decays:

- η(t) grow if Unstable- η(t) decay if Stable• Or in other word:- (+ve) slope Unstable.- (-ve) slope Stable.

*)(x

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Linear stability analysis limitation

• The linearization method cannot be used :

- If f(x) has a vertical tangent at F.P.- If f(x) has a horizontal tangent at F.P.

In both cases f(x) cannot be expanded in a Taylor series near x = x* since there is no Taylor series.

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Potentials

• In term of potential energy 1D system is given by:

• Potentials always decreases along trajectories Particle always moves toward lower potentials.

• Local minima of V(x) Stable F.P• Local maxima of V(x) Unstable F.P

dx

dV

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Notes on 1D line systems

• If f(x) is smooth enough, then the solution exist and unique.

• The approach to equilibrium always monotonic.

• If we flow monotonically on a line, we will never come back to starting point, so oscillations can never occur. (there are no periodic solutions)

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Bifurcations in one-dimensionalsystems

• Bifurcation theory is the study of how solutions of dynamical systems change as a parameter is changed.

• For a dependence parameter T:If the phase portraits for T < Tc and T > Tc are qualitatively different, we say that a bifurcation occurs at T = Tc.

• The qualitative change lead to create or destroy of F.Ps or change in there stability.

xfx

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Bifurcation

Bifurcation

Static

Saddle-Node Transcritical Pitchfork

Dynamic

Hopf

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The saddle-node bifurcation• As the r varied stable and unstable F.Ps meet each

other and both of them destroyed or vanish.

• There are three different possibilities:(a) , there are two distinct fixed points:(b) , there's only one fixed point: (c) , there’s no fixed points (F.P cannot be imaginary)

2xx

22 0 xx

0 *x

0 0*x

0

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The saddle-node bifurcation

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The Transcritical bifurcation • Two solutions stable and unstable when they

met they exchange their stability stable one become unstable and the unstable one become stable.

• If α < 0 :- x* = 0 ----> Stable.- x *= α ----> Unstable.• If α > 0:- x* = 0 ----->Unstable- x *= α ----> Stable

2xxx

*,0*02 xxxx

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The supercritical pitchfork bifurcation

• Common in physical problems that have symmetry. F.P tend to appear and disappear in symmetrical pairs.

• There are three different possibilities:(a) , (F.P cannot be imaginary) , so there's

only one fixed point: Stable(b) , one fixed point: Stable(c) , there are three fixed points:

3xxx

*,0* xx

0

00

0*x

0*x

0*x Unstable *x Stable

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The subcritical pitchfork bifurcation

• This kind of bifurcation is dangerous. • There are three different possibilities:(a) , (F.P cannot be imaginary), only one

fixed point: Unstable(b) , only one fixed point: Unstable(c) , there are three fixed points:

3xxx *,0* xx

0

00

0*x

0*x

0*x

Unstable *x

Stable

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Flows on the circle

• The main difference with flows on the line is that now the flow can return to where it was, by going around the circle. Thus, periodic solutions are possible!

• flow CCW.

• flow CW.

0

0

)( f

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The uniform oscillator

• The simplest flow on the circle is given by the system:

• This system is easily solved:

• All points are back where they started after period of:

0 t

2

T

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The non-uniform oscillator

• More complicated system:• Fixed points:• There are three different cases:- No Solution. The whole circle rotates towards

increasing angles. By integrating the governing equation we get the

period to be:

,sin a

sina

0, a

a

22

2

aT

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The non-uniform oscillator

• , one fixed point ---> ,half-stable.

• , there are two distinct fixed points, one stable and one unstable. These are born out of a saddle-node bifurcation at the critical value

a 2*

a

a

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Two-dimensional linear systems

Ex: Spring mass system

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Stability Language

• If F.P attracts all trajectories in the phase plane it is called global attracting.

• If all trajectories that start sufficiently close to F.P remains close to it for all time , the F.P is Liapunov stable.

• If nearby trajectories are neither attracted nor repelled from F.P, then F.P is neutrally stable (Center).

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Classification of linear system

For the 2 x 2 matrix A

The characteristic equation:

Eigen-values:

Where:

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Attracting directions

• Trajectories approach the F.P tangent to slow Eigen-direction , (Eigen-vector with the smallest |λ|), and parallel to the fast Eigen-direction.

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Some special cases

• λ2< λ1 or λ1< λ2 : case (a) & (b).• λ1= λ2= λ: Star node: (c)• λ1= λ2= 0: infinite F.P,• λ 1 or λ2= 0: (d)• One Eigen-vector: Degenerate: (e)

0x

(a) (b) (c) (d) (e)

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Classification of 2D linear system

• Unstable Saddle point.

• : Three cases:

00

042 0

0

Stable

UnstableNode

042 Star

Degenerate Node

Spiral

Center

042

0

0

Stable

Unstable

0 Neut. Stable0]Re[

0]Re[

0]Re[

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Classification of linear system

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Nonlinear systems in the phase plane

2122

2111

,

,

xxfx

xxfx

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Linearization, Jacobean matrix

• The stability of a fixed point is determined by the Eigen-values and Eigen-vectors of the Jacobean evaluated at the fixed point

2

2

1

2

2

1

1

1

x

f

x

fx

f

x

f

J

),(* *2

*1 xxx

sEigenvalueIJ 0||

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Classifications of the fixed points

• Robust cases

• Marginal cases

Unstable

Unstable

Stable

saddle

source

k

ddistribute

repellers

attractors

sin

,

0]Re[

0]Re[

StableCenterimaginary 0]Re[

At least one λ is zero -----> high order & non-isolated fixed point

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Definitions

• Hyperbolic F.P:

If • Structural stability phase portrait:

If the topology of the phase portrait cannot be changed by arbitrary small perturbation to the vector field. ( under damping Saddle is structurally stable , while center is not , convert to spiral).

• Basin of attraction: It’s the set of IC’s such that: x(t) x* as t ∞.

0]Re[ s

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Definitions

• Basin boundary: The line separates the basins for two nodes.• Separatix: The line or orbit separates regions in the phase

space corresponds to qualitatively different dynamical behavior. (separates stable and unstable)

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Conservative systems• This name is used for systems that originate from Newton's

law with a conservative force.

• By integrating this system we will get the following :

Where:

is the kinetic energy . V (x) is the potential energy. E is the total energy.

dx

dVxm

ExVxm )(2

1 2

2

2

1xm

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Conservative systems

• Rearrange the last equation to get the following form:

• The above eqn. can be used to plot the phase portrait manually:

- Plot V(x) versus x.- Plot several levels of E on V(x) plot.- Draw the difference (E-V) on x-x’ plane

)(2xVE

mx

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Conservative systems

• If x* is a local minimum of V(x), then (x*, 0) is a local minimum of E.

• If x* is a local maximum of V(x), then (x*, 0) is a saddle.

• If (x*, y* = 0) is an isolated F.P of the conservative system, and if it’s is local minimum of E, then (x*, y *= 0) is a center

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Pendulum 0sin L

g

sinL

g

E

L

g cos

2

1 2

first-order form Total energy

Fixed points

)0,(

)0,(*

3

2

P

P

],[

)0,0(0* 1 P

Jacobean

0cos

10

L

gJ

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PendulumCenterLgi

L

gJ

/0

10)0,0( 2,1

SaddleLgL

gJ

/0

10)0,( 2,1

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Pendulums and their phase portraits.

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Damped Pendulum

• By adding damping the system now is not conserved and from stability view Centers become spiral and Saddle remains Saddle

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Periodic Solution

• Unlike equilibrium solution, it’s a dynamic solution that characterized by time varying states and one basic frequency.

• For a continuous time system: is a periodic solution with T is the

least period if:)(tXx

TtXtX

tXTtX

0)()(

)()(

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Autonomous System (A.S)

• x: n-dimensional state vector.• M: m- dimensional parameter vector.• A periodic solution X with least T corresponds

to close orbit in the domain.• A periodic solution for A.S can be treated as a

F.P of a defined map called the Poincare map.

);( MxFx

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Definitions in A.S

• A periodic solution is called Limit cycle if there are no other periodic solutions sufficiently close to it.

• Isolated: there are no other trajectories near it, nearby trajectories either approach it (stable) or go away from it (unstable).

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Bendixson’s criterion, 2D

• For the A.S if

- Does not change sign, or does not vanish, then: The periodic solutions are not possible in D.

No chance to change signVanish for

2122

2111

,

,

xxfx

xxfx

2

2

1

1

x

f

x

f

23112

21

2 xxxx

xx

2

2

2

1

1

x

f

x

f

Periodic0

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Nonautonomus Systems (N.A.S)

• A periodic solution X with least T corresponds to close orbit in the domain.

• We can see that the major difference between the A.S and the N.A.S is that the last one depends on time explicitly.

);,( MtxFx

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Floquet Theory A.S

• For the periodic solution of A.S: • is nxn constant matrix and called Monodromy

matrix. It maps an initial vector at t=0 to another vector at t=T.

• The Eigen-values (λ) of the Mondromy matrix are called Floquet multipliers. It will be used to classify stability.

• One of the Floquet multipliers of A.S is always Unity.

)()( tYTtY

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A.S Stability

• If only one λ =1 -----> Hyperbolic .• If more than one λ =1 -----> NonHyperbolic .• Hyperbolic:- If all λ < 1 -----> Stable (Attractors).- If one or more λ >1 -----> Unstable - If all λ (other than unity) >1 -----> Unstable periodic

reppeller.- If λ distributed ( <1 & >1) -----> Unstable limit cycle

of saddle type.

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A.S Stability

• Nonhyperbolic:

- If one or more λ >1 ----> Unstable.

- If all λ <1 -----> nonlinear analysis needed

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N.A.S Stability

• In contrast with the A.S case for which one of the Floquet multipliers is always unity, in the N.A.S such conditions is not satisfied.

• If non λ lie on the unit circle (λ ≠1) -----> Hyperbolic .

• Otherwise -----> NonHyperbolic.

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N.A.S Stability

• Hyperbolic:- If all λ <1-----> Stable limit cycle.- If one or more λ >1 -----> Unstable - If all λ >1 -----> Unstable periodic reppeller.- If λ distributed ( <1 & >1) unit circle ---->

Unstable saddle type.

• NonHyperbolic: Nonlinear analysis is needed.

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Poincare Maps

• Poincare Section: It’s a hypersurface in the state space that is transverse to the flow of a given system of equations.

• A.S: N.A.S:

- n(x) is a vector normal to the section located at x.- F(x) is the vector field describing the flow.

0)()(

0)().(

xFxn

or

xFxn

T

0);()]([ txFtxnT

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Definitions

• Two sided section: section with different sign for intersections.

• One sided section: section with same sign for all intersections.

• Poincare Map: map of current intersection to the subsequent intersection.

)(1 mm xPx

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Poincare Map N.A.S

• Period (T) can be used to construct a Poincare section.

• To construct a Poincare section, we collect discrete points at intervals of the period T.

• If η represents a point on a section, the Poincare map P is defined by:

),,()( 00 tTtxP

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Poincare Map N.A.S

• In a neighborhood of the fixed point of the map, we have:

• Stability of the fixed point η0 can be obtained by studying the Eigen-values of the Jacobian matrix

||||)()()( 2000 vOvPDPvP

)( 0PD

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Stability analysis of N.A.S based on

• If all λ ≠1 -----> Hyperbolic.• If one or more λ =1 --> NonHyperbolic.• Hyperbolic:- If all λ <1 -----> Stable limit cycle.- If all λ >1 -----> Unstable repellers- If λ distributed ( <1 & >1) ----> Unstable

saddle type.• NonHyperbolic: Nonlinear analysis is needed.

)( 0PD

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Poincare Map A.S

• The period associated with a periodic orbit is not usually explicitly known.

• Stability can be analyzed according to the Jacobian matrix , elements on nth row and nth column are zero except the diagonal which is unity.

• Stability from J matrix constructed from the Jacobian matrix after deleting the nth row and nth column.

PD

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Stability analysis of N.A.S based on J

- If all λ <1-----> Stable limit cycle.

- If all λ >1 -----> Unstable repellers

- If λ distributed (<1 & >1) ----> Unstable saddle type.

- If one or more λ =1, while the rest <1 ----> Nonlinear analysis is needed.

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Bifurcation of Periodic solutionBifurcation occurs when Floquet multipliers leave the unit circle through:• +1:- Transcritical.- Symmetry breaking- Cyclic-fold• -1:- Period-doupling• Real axis:- Secondary Hopf

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Symmetry-breaking Bifurcation

• Break the symmetry of the periodic solution, similar to pitchfork.

• Supercritical (a): stable asymmetric coexist with unstable symmetric on one side of the bifurcation point.

• Subcritical (b): unstable asymmetric coexist with stable symmetric on one side of the bifurcation point.

(a) (b)

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Cyclic-fold

• A branch of stable periodic solutions and a branch of unstable periodic solutions coalesce and obliterate each other at the bifurcation point.

• Similar to the saddle node bifurcation of the fixed point.

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Period-doubling Bifurcation

• Branch of stable periodic solutions that exists before the bifurcation continues as an unstable branch of periodic solutions after the bifurcation.

• Supercritical: Branch of stable-doubled periodic solutions created.

• Subcritical: Branch of unstable-doubled periodic solutions destroyed.

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Period-doubling Bifurcation

• In the subcritical case,- unstable (2T) collides stable (T/2----> unstable

(lower period T/2).

• After k successive period-doubling bifurcations, we would have 2k points on the corresponding Poincare section.

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Transcritical Bifurcation

• Stable and unstable periodic solutions meet and exchange their stability after bifurcation point.

• Similar to Transcritical in F.P.

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Secondary Hopf or Neimark

• Stable prior to the bifurcation continues as unstable after the bifurcation.

• Supercritical: A branch of stable quasiperiodic solutions is created.

• Subcritical: A branch of stable quasiperiodic solutions is destroyed.

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Thank You