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Delay differential equation Stability charts An experiment Future work
Dynamics of delay differential equations with
distributed delays
Kiss, Gabor
BCAM - Basque Center for Applied MathematicsBilbaoSpain
February 1, 2011
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Outline
Delay differential equation
Stability charts
An experiment
Future work
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Applications
◮ Population dynamics
◮ Infections diseases
◮ Neuronal dynamics
◮ Car traffic dynamics
◮ Laser dynamics
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Wright’s equation
x(t) = −αx(t − 1){1 + x(t)} (α > 0). (1)
E. M. Wright.A non-linear difference-differential equation.J. Reine Angew. Math., 194:66–87, 1955.
J.-P. Lessard.Recent advances about the uniqueness of the slowly oscillatingperiodic solutions of Wright’s equation.J. Differential Equations, 248(5):992–1016, 2010.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Wright’s equation
x(t) = −αx(t − 1){1 + x(t)} (α > 0). (1)
E. M. Wright.A non-linear difference-differential equation.J. Reine Angew. Math., 194:66–87, 1955.
J.-P. Lessard.Recent advances about the uniqueness of the slowly oscillatingperiodic solutions of Wright’s equation.J. Differential Equations, 248(5):992–1016, 2010.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Wright’s equation
x(t) = −αx(t − 1){1 + x(t)} (α > 0). (1)
E. M. Wright.A non-linear difference-differential equation.J. Reine Angew. Math., 194:66–87, 1955.
J.-P. Lessard.Recent advances about the uniqueness of the slowly oscillatingperiodic solutions of Wright’s equation.J. Differential Equations, 248(5):992–1016, 2010.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
M. C. Mackey and L. Glass.Oscillation and chaos in physiological control systems.Science, 197(4300):287–289, 1977.
x(t) = βx(t − τ)
1 + xn(t − τ)− γx , γ, β, n > 0. (2)
G. Rost and J. Wu.Domain-decomposition method for the global dynamics ofdelay differential equations with unimodal feedback.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,463(2086):2655–2669, 2007.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
M. C. Mackey and L. Glass.Oscillation and chaos in physiological control systems.Science, 197(4300):287–289, 1977.
x(t) = βx(t − τ)
1 + xn(t − τ)− γx , γ, β, n > 0. (2)
G. Rost and J. Wu.Domain-decomposition method for the global dynamics ofdelay differential equations with unimodal feedback.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,463(2086):2655–2669, 2007.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
M. C. Mackey and L. Glass.Oscillation and chaos in physiological control systems.Science, 197(4300):287–289, 1977.
x(t) = βx(t − τ)
1 + xn(t − τ)− γx , γ, β, n > 0. (2)
G. Rost and J. Wu.Domain-decomposition method for the global dynamics ofdelay differential equations with unimodal feedback.Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,463(2086):2655–2669, 2007.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
W. Gurney, S. Blythe, and R. Nisbet.Nicholson’s blowflies revisited.Nature, 287:17–21, 1980.
Nicholson’s blowflies equation; it is of the form
x(t) = −γx(t) + px(t − τ)e−ax(t−τ) (3)
A. Nicholson.The self-adjustment of populations to change.Cold Spring Harbor Symposia on Quantitative Biology, 22:153,1957.
A. Nicholson.An outline of the dynamics of animal populations.Insect ecology and population management: readings intheory, technique, and strategy, 2:3, 1972.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
W. Gurney, S. Blythe, and R. Nisbet.Nicholson’s blowflies revisited.Nature, 287:17–21, 1980.
Nicholson’s blowflies equation; it is of the form
x(t) = −γx(t) + px(t − τ)e−ax(t−τ) (3)
A. Nicholson.The self-adjustment of populations to change.Cold Spring Harbor Symposia on Quantitative Biology, 22:153,1957.
A. Nicholson.An outline of the dynamics of animal populations.Insect ecology and population management: readings intheory, technique, and strategy, 2:3, 1972.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
x(t) = −ax(t)− bg(x(t − τ)), a, b ∈ R, τ ≥ 0 (4)
x(t) = −ax(t) + g
(∫
h
0x(t − τ)dµ(τ)
)
. (5)
Phase space: the Banach C = C ([0, h],R) space of continuousfunctions mapping the interval [0, h] into R, with the supremumnorm. Here a ∈ R, g ∈ C 1 and the integral is of Stieltjes-type.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
x(t) = −ax(t)− bg(x(t − τ)), a, b ∈ R, τ ≥ 0 (4)
x(t) = −ax(t) + g
(∫
h
0x(t − τ)dµ(τ)
)
. (5)
Phase space: the Banach C = C ([0, h],R) space of continuousfunctions mapping the interval [0, h] into R, with the supremumnorm. Here a ∈ R, g ∈ C 1 and the integral is of Stieltjes-type.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Typically, only one time lag has been introduced inmodeling using differential–delay equations, but forbetter models and for mathematical interest it isdesirable to study equations in which two or more ormore time–lags may appear.
R. D. Nussbaum.Differential-delay equations with two time lags.Mem. Amer. Math. Soc., 16(205):vi+62, 1978.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
The integral is of Stieltjes-type, µ : R → R is a non–decreasingand right-continuous function satisfying
(A1) µ(τ) = 1, if τ ≥ h
and
(A2) µ(τ) = 0, if τ < 0,
where a, b ∈ R h ≥ 0. (A1) and (A2) together with monotonicityof function µ imply that
∫
h
0dµ(τ) = 1. (6)
In (8) E is the E =∫
h
0 τdµ(τ) average delay.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
τ
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
If xt is an equilibrium point of (5) then yψt = D2F (t, x)ψ is theunique solution of the linear variational equation which, when (5)is considered, is of form
y(t) = −ay(t)− b
∫
h
0y(t + τ)dµ(τ) (7)
where b = −g ′(x), x ∈ C , x ∈ R.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
x(t) = −ax(t)− bx(t − E ) (8)
and
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ). (9)
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
TheoremThe zero solution x ≡ 0 of x(t) = −ax(t)− bx(t − E ) isasymptotically stable if
E <arccos
(
− a
b
)
√b2 − a2
, b > |a|.
0
2
4
1
3
5
Γ0
Γ+
1
Γ+
2
Γ−
1
Γ−
2
a
b
Figure: Stability charts of x(t) = −ax(t)− bx(t − E ) for E = 1
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
TheoremThe zero solution x ≡ 0 of x(t) = −ax(t)− bx(t − E ) isasymptotically stable if
E <arccos
(
− a
b
)
√b2 − a2
, b > |a|.
0
2
4
1
3
5
Γ0
Γ+
1
Γ+
2
Γ−
1
Γ−
2
a
b
Figure: Stability charts of x(t) = −ax(t)− bx(t − E ) for E = 1
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Theorem (Krisztin)
The zero solution x ≡ 0 of x(t) = −b∫
h
0 x(t − τ)dµ(τ), isasymptotically stable if
E =
∫
h
0τdµ(τ) <
π
2b.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
The corresponding function and equation related to
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ), (10)
are
h(λ) : C → C, λ 7→ λ+ a + b
∫
h
0e−λτdµ(τ) (11)
and
λ+ a + b
∫
h
0e−λτdµ(τ) = 0, λ ∈ C. (12)
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
DefinitionLet µ : R → R be a monotonically nondecreasing function withexpected value E . We say that µ is symmetric about itsexpectation if
µ(E − x) = 1− µ(E + x − 0). (13)
LemmaLet µ : R → R be symmetric about its expectation E > 0 in
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ). (14)
Then Γ0 ≺ |Γ+k,l | and Γ0 ≺ |Γ−
k,m| on I+k
and I−k, respectively, for
1 ≤ l ≤ i , 1 ≤ m ≤ j .
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
DefinitionLet µ : R → R be a monotonically nondecreasing function withexpected value E . We say that µ is symmetric about itsexpectation if
µ(E − x) = 1− µ(E + x − 0). (13)
LemmaLet µ : R → R be symmetric about its expectation E > 0 in
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ). (14)
Then Γ0 ≺ |Γ+k,l | and Γ0 ≺ |Γ−
k,m| on I+k
and I−k, respectively, for
1 ≤ l ≤ i , 1 ≤ m ≤ j .
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
TheoremLet us fix a number E > 0, furthermore, let b > |a|, and consider
x(t) = −ax(t)− bx(t − E ). (15)
Let us suppose that the trivial solution x ≡ 0 of equation (15) isasymptotically stable for a given pair of parameters a, b. Then thetrivial solution x ≡ 0 of equation
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ), (16)
given with an arbitrary distribution function that is symmetricabout the fixed expectation E is asymptotically stable.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
G. Kiss and B. KrauskopfStability implications of delay distribution for first-order andsecond-order systems.Discrete and Continuous Dynamical Systems - Series B,13:327:345, 2010.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
TheoremLet µ be symmetric about its expected value E. Then, the trivialsolution x = 0 of
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ), (17)
is asymptotically stable if
E <arccos(−a
b)√
b2 − a2, where b > |a|.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Does delay distribution always increase the stability region?
x(t) = −x(t)− ax(t)− bx(t − 1), (18)
and
x(t) = −x(t)− ax(t)− b
(
1
2x(t − τ1) +
1
2x(t − τ2)
)
, (19)
where τ1 = 0.55 and τ1 = 1.45, so that we have a mean of E = 1
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Does delay distribution always increase the stability region?
x(t) = −x(t)− ax(t)− bx(t − 1), (18)
and
x(t) = −x(t)− ax(t)− b
(
1
2x(t − τ1) +
1
2x(t − τ2)
)
, (19)
where τ1 = 0.55 and τ1 = 1.45, so that we have a mean of E = 1
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
-60
-40
-20
0
20
40
60
0 20 40 60 80 100 120 140a
b
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Is stability preserving order dependent?
x(t) = −ax(t)− bx(t − E ).
and
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ),
G. Kiss and B. KrauskopfStabilizing effect of delay distribution for a class ofsecond-order systems without instantaneous feedback.Dynamical Systems, 2010., In Press
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Is stability preserving order dependent?
x(t) = −ax(t)− bx(t − E ).
and
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ),
G. Kiss and B. KrauskopfStabilizing effect of delay distribution for a class ofsecond-order systems without instantaneous feedback.Dynamical Systems, 2010., In Press
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Is stability preserving order dependent?
x(t) = −ax(t)− bx(t − E ).
and
x(t) = −ax(t)− b
∫
h
0x(t − τ)dµ(τ),
G. Kiss and B. KrauskopfStabilizing effect of delay distribution for a class ofsecond-order systems without instantaneous feedback.Dynamical Systems, 2010., In Press
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35))
-15
-10
-5
0
5
10
15
-10 -5 0 5 10
b
a
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35))
4.65
4.7
4.75
4.8
4.85
4.9
0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .48, b = 4.85
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 1000 2000 3000 4000 5000 6000
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .48, b = 4.85
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
4551 4552 4553 4554 4555
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .48, b = 4.85
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0.145 0.15 0.155 0.16 0.165 0.17 0.175
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .4, b = 4.85
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 1000 2000 3000 4000 5000 6000
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .4, b = 4.85
-0.4
-0.2
0
0.2
0.4
5505 5510 5515 5520 5525
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .4, b = 4.85
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
a = .4, b = 4.85
0.1
0.15
0.2
0.25
0.3
0.1 0.15 0.2 0.25 0.3
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Equations with two delays
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},a = 0, b = 4.85
-1
-0.5
0
0.5
1
1.5
2
2.5
250 252 254 256 258 260
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
TheoremLet a = 0. Then
x(t) = −ax(t)− b (0.5x(t − 1.65) + 0.5x(t − 0.35)) {1 + x(t)},
has at least three nontrivial coexisting periodic solutions at theparameter value b = 6.8.
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Computation needed
◮ Periodic solutions to the Van der Pole’s oscillator
x(t)− εx(t)(1− x2(t)) + x(t − τ)− kx(t) = 0. (20)
R. D. NussbaumPeriodic solutions of some nonlinear autonomousfunctional differential equation.Ann. Mat. Pura Appl. (4), 101:263–306, 1974.
◮ Compute the stability of periodic solutions
◮ Compute invariant tori in infinite dimension
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Computation needed
◮ Periodic solutions to the Van der Pole’s oscillator
x(t)− εx(t)(1− x2(t)) + x(t − τ)− kx(t) = 0. (20)
R. D. NussbaumPeriodic solutions of some nonlinear autonomousfunctional differential equation.Ann. Mat. Pura Appl. (4), 101:263–306, 1974.
◮ Compute the stability of periodic solutions
◮ Compute invariant tori in infinite dimension
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged
Delay differential equation Stability charts An experiment Future work
Computation needed
◮ Periodic solutions to the Van der Pole’s oscillator
x(t)− εx(t)(1− x2(t)) + x(t − τ)− kx(t) = 0. (20)
R. D. NussbaumPeriodic solutions of some nonlinear autonomousfunctional differential equation.Ann. Mat. Pura Appl. (4), 101:263–306, 1974.
◮ Compute the stability of periodic solutions
◮ Compute invariant tori in infinite dimension
Dynamics of delay differential equations with distributed delays BCAM and University of Szeged