Dynamics of delay differential equations with distributed ... · PDF fileDelay differential equation Stability charts An experiment Future work Dynamics of delay differential equations

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


Outline

Delay differential equation

Stability charts

An experiment

Future work



Applications

◮ Population dynamics

◮ Infections diseases

◮ Neuronal dynamics

◮ Car traffic dynamics

◮ Laser dynamics



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.



Wright’s equation

x(t) = −αx(t − 1){1 + x(t)} (α > 0). (1)





Wright’s equation

x(t) = −αx(t − 1){1 + x(t)} (α > 0). (1)





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.




x(t) = βx(t − τ)

1 + xn(t − τ)− γx , γ, β, n > 0. (2)





x(t) = βx(t − τ)

1 + xn(t − τ)− γx , γ, β, n > 0. (2)




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.



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.



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.



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.



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.



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.



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



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.



x(t) = −ax(t)− bx(t − E ) (8)

and

x(t) = −ax(t)− b

∫

h

0x(t − τ)dµ(τ). (9)



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



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



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.



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)



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 .



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 .



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.



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.



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|.



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



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



-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120 140a

b



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




x(t) = −ax(t)− bx(t − E ).

and

x(t) = −ax(t)− b

∫

h






x(t) = −ax(t)− bx(t − E ).

and

x(t) = −ax(t)− b

∫

h





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




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




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




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




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




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




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




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




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




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



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.



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



Computation needed


x(t)− εx(t)(1− x2(t)) + x(t − τ)− kx(t) = 0. (20)






Computation needed


x(t)− εx(t)(1− x2(t)) + x(t − τ)− kx(t) = 0. (20)





Documents

Dynamics of delay differential equations with distributed ... · PDF fileDelay differential equation Stability charts An experiment Future work Dynamics of delay differential equations