Dynamics of delay differential equations with distributed delays

Transcription

1 Dynamics of delay differential equations with distributed delays Kiss, Gábor BCAM - Basque Center for Applied Mathematics Bilbao Spain February 1, 211

2 Outline Delay differential equation Stability charts An experiment Future work

3 Applications Population dynamics Infections diseases Neuronal dynamics Car traffic dynamics Laser dynamics

4 Wright s equation ẋ(t) = αx(t 1){1+x(t)} (α > ). (1) E. M. Wright. A non-linear difference-differential equation. J. Reine Angew. Math., 194:66 87, J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright s equation. J. Differential Equations, 248(5): , 21.

5 Wright s equation ẋ(t) = αx(t 1){1+x(t)} (α > ). (1) E. M. Wright. A non-linear difference-differential equation. J. Reine Angew. Math., 194:66 87, J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright s equation. J. Differential Equations, 248(5): , 21.

6 Wright s equation ẋ(t) = αx(t 1){1+x(t)} (α > ). (1) E. M. Wright. A non-linear difference-differential equation. J. Reine Angew. Math., 194:66 87, J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright s equation. J. Differential Equations, 248(5): , 21.

7 M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197(43): , x(t τ) ẋ(t) = β 1+x n γx, γ,β,n >. (2) (t τ) G. Röst and J. Wu. Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463(286): , 27.

8 M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197(43): , x(t τ) ẋ(t) = β 1+x n γx, γ,β,n >. (2) (t τ) G. Röst and J. Wu. Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463(286): , 27.

9 M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197(43): , x(t τ) ẋ(t) = β 1+x n γx, γ,β,n >. (2) (t τ) G. Röst and J. Wu. Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463(286): , 27.

10 W. Gurney, S. Blythe, and R. Nisbet. Nicholson s blowflies revisited. Nature, 287:17 21, 198. Nicholson s blowflies equation; it is of the form ẋ(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, A. Nicholson. An outline of the dynamics of animal populations. Insect ecology and population management: readings in theory, technique, and strategy, 2:3, 1972.

11 W. Gurney, S. Blythe, and R. Nisbet. Nicholson s blowflies revisited. Nature, 287:17 21, 198. Nicholson s blowflies equation; it is of the form ẋ(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, A. Nicholson. An outline of the dynamics of animal populations. Insect ecology and population management: readings in theory, technique, and strategy, 2:3, 1972.

12 ẋ(t) = ax(t) bg(x(t τ)), a,b R, τ (4) ( h ẋ(t) = ax(t)+g ) x(t τ)dµ(τ). (5) Phase space: the Banach C = C([,h],R) space of continuous functions mapping the interval [, h] into R, with the supremum norm. Here a R, g C 1 and the integral is of Stieltjes-type.

13 ẋ(t) = ax(t) bg(x(t τ)), a,b R, τ (4) ( h ẋ(t) = ax(t)+g ) x(t τ)dµ(τ). (5) Phase space: the Banach C = C([,h],R) space of continuous functions mapping the interval [, h] into R, with the supremum norm. Here a R, g C 1 and the integral is of Stieltjes-type.

14 Typically, only one time lag has been introduced in modeling using differential delay equations, but for better models and for mathematical interest it is desirable to study equations in which two or more or more time lags may appear. R. D. Nussbaum. Differential-delay equations with two time lags. Mem. Amer. Math. Soc., 16(25):vi+62, 1978.

15 The integral is of Stieltjes-type, µ : R R is a non decreasing and right-continuous function satisfying (A1) µ(τ) = 1, if τ h and (A2) µ(τ) =, if τ <, where a,b R h. (A1) and (A2) together with monotonicity of function µ imply that h In (8) E is the E = h τdµ(τ) average delay. dµ(τ) = 1. (6)

16 τ

17 If ˆx t is an equilibrium point of (5) then y ψ t = D 2 F(t,ˆx)ψ is the unique solution of the linear variational equation which, when (5) is considered, is of form ẏ(t) = ay(t) b where b = g (x), ˆx C,x R. h y(t +τ)dµ(τ) (7)

18 ẋ(t) = ax(t) bx(t E) (8) and ẋ(t) = ax(t) b h x(t τ)dµ(τ). (9)

19 Theorem The zero solution x of ẋ(t) = ax(t) bx(t E) is asymptotically stable if ) E < arccos( a b,b > a. b 2 a2 b 4 Γ + 2 Γ Γ 1 3 Γ 1 5 Γ 2 a Figure: Stability charts of ẋ(t) = ax(t) bx(t E) for E = 1

20 Theorem The zero solution x of ẋ(t) = ax(t) bx(t E) is asymptotically stable if ) E < arccos( a b,b > a. b 2 a2 b 4 Γ + 2 Γ Γ 1 3 Γ 1 5 Γ 2 a Figure: Stability charts of ẋ(t) = ax(t) bx(t E) for E = 1

21 Theorem (Krisztin) The zero solution x of ẋ(t) = b h x(t τ)dµ(τ), is asymptotically stable if E = h τdµ(τ) < π 2b.

22 The corresponding function and equation related to are and ẋ(t) = ax(t) b h h(λ) : C C, λ λ+a+b λ+a+b h x(t τ)dµ(τ), (1) h e λτ dµ(τ) (11) e λτ dµ(τ) =, λ C. (12)

23 Definition Let µ : R R be a monotonically nondecreasing function with expected value E. We say that µ is symmetric about its expectation if µ(e x) = 1 µ(e +x ). (13) Lemma Let µ : R R be symmetric about its expectation E > in ẋ(t) = ax(t) b h x(t τ)dµ(τ). (14) Then Γ Γ + k,l and Γ Γ k,m on Ĩ+ k and Ĩ k 1 l i, 1 m j., respectively, for

24 Definition Let µ : R R be a monotonically nondecreasing function with expected value E. We say that µ is symmetric about its expectation if µ(e x) = 1 µ(e +x ). (13) Lemma Let µ : R R be symmetric about its expectation E > in ẋ(t) = ax(t) b h x(t τ)dµ(τ). (14) Then Γ Γ + k,l and Γ Γ k,m on Ĩ+ k and Ĩ k 1 l i, 1 m j., respectively, for

25 Theorem Let us fix a number E >, furthermore, let b > a, and consider ẋ(t) = ax(t) bx(t E). (15) Let us suppose that the trivial solution x of equation (15) is asymptotically stable for a given pair of parameters a, b. Then the trivial solution x of equation ẋ(t) = ax(t) b h x(t τ)dµ(τ), (16) given with an arbitrary distribution function that is symmetric about the fixed expectation E is asymptotically stable.

26 G. Kiss and B. Krauskopf Stability implications of delay distribution for first-order and second-order systems. Discrete and Continuous Dynamical Systems - Series B, 13:327:345, 21.

27 Theorem Let µ be symmetric about its expected value E. Then, the trivial solution x = of ẋ(t) = ax(t) b is asymptotically stable if h E < arccos( a b ), where b > a. b 2 a2 x(t τ)dµ(τ), (17)

28 Does delay distribution always increase the stability region? and ẍ(t) = ẋ(t) ax(t) bx(t 1), (18) ẍ(t) = ẋ(t) ax(t) b ( 1 2 x(t τ 1)+ 1 ) 2 x(t τ 2), (19) where τ 1 =.55 and τ 1 = 1.45, so that we have a mean of E = 1

29 Does delay distribution always increase the stability region? and ẍ(t) = ẋ(t) ax(t) bx(t 1), (18) ẍ(t) = ẋ(t) ax(t) b ( 1 2 x(t τ 1)+ 1 ) 2 x(t τ 2), (19) where τ 1 =.55 and τ 1 = 1.45, so that we have a mean of E = 1

30 b a

31 Is stability preserving order dependent? ẍ(t) = aẋ(t) bx(t E). and ẍ(t) = aẋ(t) b h x(t τ)dµ(τ), G. Kiss and B. Krauskopf Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback. Dynamical Systems, 21., In Press

32 Is stability preserving order dependent? ẍ(t) = aẋ(t) bx(t E). and ẍ(t) = aẋ(t) b h x(t τ)dµ(τ), G. Kiss and B. Krauskopf Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback. Dynamical Systems, 21., In Press

33 Is stability preserving order dependent? ẍ(t) = aẋ(t) bx(t E). and ẍ(t) = aẋ(t) b h x(t τ)dµ(τ), G. Kiss and B. Krauskopf Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedback. Dynamical Systems, 21., In Press

34 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)) b a

35 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35))

36 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.48, b =

37 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.48, b =

38 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.48, b =

39 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.4, b =

40 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.4, b =

41 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.4, b =

42 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =.4, b =

43 Equations with two delays ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, a =, b =

44 Theorem Let a =. Then ẋ(t) = ax(t) b(.5x(t 1.65) +.5x(t.35)){1+x(t)}, has at least three nontrivial coexisting periodic solutions at the parameter value b = 6.8.

45 Computation needed Periodic solutions to the Van der Pole s oscillator ẍ(t) εẋ(t)(1 x 2 (t))+x(t τ) kx(t) =. (2) R. D. Nussbaum Periodic solutions of some nonlinear autonomous functional differential equation. Ann. Mat. Pura Appl. (4), 11:263 36, Compute the stability of periodic solutions Compute invariant tori in infinite dimension

46 Computation needed Periodic solutions to the Van der Pole s oscillator ẍ(t) εẋ(t)(1 x 2 (t))+x(t τ) kx(t) =. (2) R. D. Nussbaum Periodic solutions of some nonlinear autonomous functional differential equation. Ann. Mat. Pura Appl. (4), 11:263 36, Compute the stability of periodic solutions Compute invariant tori in infinite dimension

47 Computation needed Periodic solutions to the Van der Pole s oscillator ẍ(t) εẋ(t)(1 x 2 (t))+x(t τ) kx(t) =. (2) R. D. Nussbaum Periodic solutions of some nonlinear autonomous functional differential equation. Ann. Mat. Pura Appl. (4), 11:263 36, Compute the stability of periodic solutions Compute invariant tori in infinite dimension