A state-dependent model with unimodal feedback Qingwen Hu Center - - PowerPoint PPT Presentation

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A state-dependent model with unimodal feedback

Qingwen Hu

Center for Nonlinear Analysis Department of Mathematical Sciences The University of Texas at Dallas Richardson, Texas

October 2014, UTD

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Examples

Bellman’s Questions (Book: Differential -difference equations (1963), page 80): . .

1

Under what conditions does u′(t) = au(t −u(t)), have a unique solution? . .

2

Under what conditions does u′(t) = 1 2 + 1 2u(t)−u(t −u(t)), have a unique solution?

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Driver’s Existence and Uniqueness Results:

Consider the system { ˙ y(t) = f(y(t), y(t −z(t))), ˙ z(t) = g(y(t), y(t −z(t)), z(t)) (1) where (y(t), z(t)) ∈ Rn ×R. Let D be an open set in Rn ×Rn and z0 ≤ ¯ r be positive constants. Assume that (i) f : D ∋ (θ1, θ2) → f(θ1, θ2) ∈ Rn and g : D×R ∋ (θ1, θ2, θ3) → g(θ1, θ2, θ3) ∈ R are continuous and each is Lipschitz continuous with respect to θ1; (ii ) g(θ1, θ2, θ3) < 1 for all (θ1, θ2, θ3) ∈ D×R; (iii) There is a continuous map ϕ : [−¯ r, 0] → Rn with (ϕ(0), ϕ(−z0)) ∈ D. Then system (1) with given initial conditions z(0) = z0 and y(t) = ϕ(t) for t ∈ [−¯ r, 0] has a unique solution on [0, β) for some β > 0.

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Rekhlitskii’s problem: (Dokl, Akad. Nauk SSSR, vol 118, 1958, 447–449)

Consider the scalar equation u′(t)+λu(t −a(t)) = v(t), t ≥ 0 u(t) = ϕ(t), t ≤ 0, a(t) ≥ 0. If a(t) → a0 > 0 as t → +∞, under which conditions are all solutions of the equation bounded as t → +∞?

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The model

We consider attractivity of the equilibrium states of the following model { ˙ x(t) = −µx(t)+b(x(t −τ(t))) ˙ τ(t) = h(x(t), τ(t)), (SDDE) where i) µ > 0, x(t) ∈ R; ii) h is C1 (continuously differentiable); iii) The time delay τ depends on the system state (x, τ) and hence called a state-dependent delay. iv) b is C2 and is unimodal on [0, +∞) with the following properties (H1, H2, H3):

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b b b

y ξ y = µξ y = b(ξ) u ξ0 (H1) b(0) = 0 and there exists M0 > 0 or M0 = +∞ such that M0 = inf{ζ : b(ξ) > 0 for every ξ ∈ (0, ζ)}; (H2) There exists a unique ξ0 ∈ (0, M0) such that { b′(ξ) > 0 if 0 ≤ ξ < ξ0, b′(ξ) < 0 if ξ > ξ0; (2) (H3) b′′(ξ) < 0 if ξ ∈ (0, ξ0).

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Background of the model

• Fish dynamics (Arino, Lhassan, and Bravo);
• Internet conjestion model (Briat, Hjalmarsson, et al);
• Counterpart model with constant delay and with different assumptions
• n the feedback is extensively investigated in literature;
• Similar model with increasing feedback b was investigated by [Bartha

2001, Chen 2003].

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Outline

" Purposes of the work; " Existence and positivity of solutions; " Invariant intervals and attractively of solutions; " Examples; " Summary, Remarks and References.

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Purposes of the work

• Invariant order intervals;
• Positivity of the solutions;
• Attractivity of the nonnegative stationary states;
• Hopf bifurcation from nonnegative stationary states (for two examples).

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Existence and positivity of solutions

We further assume that (H4) Let u be the positive stationary state of equation (SDDE) if b′(0) > µ. There exists a constant A with A > max{u, ξ0, b(ξ0)/µ} such that b(x) < µx for all x > A; (H5) There exists a constant L > 0 such that h(x, τ) <

L L+1 for all

(x, τ) ∈ Rn+1; (H6) h(x, 0) > 0 for all x ∈ R.

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We choose r > 0 and let X0 be the set of continuous real functions φ : [−r, 0] → [0, M0] endowed with the supremum norm ∥φ∥ = max

s∈[−r,0]|φ(s)|,

where M0 is defined in (H1). Using Driver’s work on state-dependent DEs, we obtain .

Theorem

. . Assume that (H1–H6) hold. Then for every φ ∈ X0 and τ0 ∈ (0, r], there exists a unique solution (x, τ) : [0, +∞) ∋ t → (x(t), τ(t)) ∈ [0, M0]×(0, +∞), for system (SDDE) through (φ, τ0).

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Invariant sets and attactivity of stationary states

(H7) For every x ∈ R+, there exists a unique positive y := l(x) such that h(x, y) = 0. Let the set of stationary states of x-component in (SDDE) be E := {x ∈ R : b(x) = µx}. Then by (H7), every stationary state of (SDDE) can be denoted by (x, l(x)) for some x ∈ E.

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We have the following elementary results .

Lemma

. . Assume (H1–H6) hold. Let ¯ u be the maximal nonnegative stationary state of x in (SDDE). If ¯ u < ξ0, then there exists T > 0 large enough so that x(t) ∈ [0, ξ0] for all t > T. Moreover, for every solution (x, τ) of system (SDDE) with initial value (φ, τ0) ∈ X0 ×(0, r], the ω-limit set ω(φ) satisfies ω(φ)∩[0∗, ξ0∗] ̸= / 0, where we use ∗ to indicate constant functions.

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The Fluctuation Method

One important use of invariant intervals, as well as the boundedness of solutions in the current work is for the fluctuation method (Thieme, Zhao, Hsu) which has been repeatedly employed to prove existence of certain limits: Let x : [a, +∞) → R be a continuously differentiable function. If limsup

t→+∞

x(t) = x∞ < ∞, liminf

t→+∞ x(t) = x∞ < ∞ are finite with x∞ < x∞,

Then there exist sequences {tn} and {sn} in R with limn→+∞tn = limn→+∞ sn = +∞ such that    ˙ x(tn) = 0, lim

n→+∞x(tn) = x∞,

˙ x(sn) = 0, lim

n→+∞x(sn) = x∞.

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We focus on attactivity of the x-component of the solutions because of .

Lemma

. . Assume (H1–H7) hold. If limt→+∞ x(t) = c0 for some constant c0 ∈ R+, then either lim

t→+∞τ(t) = l(c0)

• r

lim

t→+∞τ(t) = +∞.

If, in addition, we have liminfτ→+∞ h(c0, τ) < 0, then lim

t→+∞τ(t) = l(c0).

Sketch of the proof: i) Use the fluctuation method to show that limt→+∞ τ(t) = +∞ if the limit limt→+∞ τ(t) does not exist; ii) Proof by contradiction for the second part.

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We distinguish three cases for the stationary states of system (SDDE):

• CaseA. b′(0) ≤ µ and (SDDE) has a unique stationary state (0, l(0));
• CaseB. b′(0) > µ and u ≤ ξ0, (SDDE) has a stationary state (0, l(0))

and a positive stationary state (u, l(u)) with b′(u) ≥ 0;

• CaseC. b′(0) > µ and u > ξ0, (SDDE) has a stationary state (0, l(0))

and a positive stationary state (u, l(u)) with b′(u) < 0.

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Case A: b′(0) ≤ µ.

.

Theorem

. . Assume (H1–H7) hold. If b′(0) ≤ µ, then we have lim

t→+∞x(t) = 0.

If, in addition, liminfτ→+∞ h(0, τ) < 0, then lim

t→+∞τ(t) = l(0).

Sketch of the proof: i) Show that {x(t) : t ∈ [0, +∞)} is bounded, using the first Lemma; ii) Use the fluctuation method to show the limit limt→+∞ x(t) exists.

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Case B: b′(0) > µ, u ≤ ξ0.

.

Theorem

. . Assume (H1–H7) hold. Suppose that the initial data ϕ(t) satisfies ϕ(t) ≥ 0 for every t ∈ [−τ0, 0] and there exists s ∈ [−τ0, 0] so that ϕ(s) > 0. If b′(0) > µ, u ≤ ξ0 and limτ→+∞ h(0, τ) exists in R∪{±∞}, then lim

t→+∞x(t) = u.

If, in addition, limτ→+∞ h(u, τ) < 0, then lim

t→+∞τ(t) = l(u).

Sketch of the proof: i) Use the fluctuation method to show limt→+∞ x(t) exists; ii) Show that the order interval [ε0∗, ξ0∗] is invariant from t = 0 for every ε ∈ (0, u), which leads to limt→+∞ x(t) = u.

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Case C: b′(0) > µ, u > ξ0.

Define ¯ b : R+ → R by ¯ b(x) := max

v∈[0,x]b(v).

Let δ > 0 be such that ¯ b(δ) = µδ. If u > ξ0, we have δ = b(ξ0)/µ and define γ = b(δ) µ = b (

b(ξ0) µ

) µ .

b b b

y ξ y = µξ y = ¯ b(ξ) u ξ0 δ γ

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The following lemma was obtained in a similar manner as in the work of [ Rost and Wu ] for the counterpart model with constant delay. .

Lemma

. . Assume (H1)–(H7). If b′(0) > µ and u > ξ0, then the order interval J := [¯ γ∗, δ∗] ⊂ X0 with ¯ γ = max{0, γ} attracts every solution whose nonnegative initial value ϕ satisfies ϕ(s) > 0 for some s ∈ [−τ0, 0].

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For Case C, we further assume that (H8) There exists a positive constant M ≥ b(ξ0)/µ so that the map R ∋ x → b(x)/x ∈ R is strictly decreasing for x ∈ (0, M] and the map b satisfies Property (P): For every v, w ∈ (0, M] with v ≤ u ≤ w, µv ≥ b(w) and µw ≤ b(v) imply that v = w.

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.

Lemma

. . [Hsu, Zhao] Either of the following two conditions is sufficient for the property (P) in condition (H8) to hold: (P1) xb(x) is strictly increasing for x ∈ (0,M]; (P2) b(x) is nonincreasing for x ∈ (u, M) and b(b(x)/µ)

x

is strictly decreasing for x ∈ (0, u].

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.

Theorem

. . Assume (H1)–(H8). Suppose that the nonnegative initial data ϕ(t) satisfies that ϕ(s) > 0 for some s ∈ [−τ0, 0]. If b′(0) > µ and u > ξ0, then we have lim

t→+∞x(t) = u.

If, in addition, liminfτ→+∞ h(u, τ) < 0, then lim

t→+∞τ(t) = l(u).

Sketch of the proof: i) Show that liminft→+∞ x(t) > 0; ii) Construct a monotone representation of b and use the fluctuation method to show that limt→+∞ x(t) > 0 exists.

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Monotone Representation

Define the map χ : (0, M]×(0, M] → R+ by χ(α, β) = { min{b(x) : α ≤ x ≤ β}if α ≤ β, max{b(x) : β ≤ x ≤ α} if β ≤ α. In [Thieme, JMB 1979], the map χ is called a monotone representation of the function b. Then i) χ is nondecreasing in α ∈ (0, M] and is nonincreasing in β ∈ (0, M]; ii) χ(x, x) = b(x) for every x ∈ (0, M] and χ is continuous on (0, M]×(0, M].

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Example I – The Ricker-Type feedback

Consider { ˙ x(t) = −µx(t)+ px(t −τ(t))e−qx(t−τ(t)), ˙ τ(t) = 1−J(x)(1+tanh(τ)), (Ricker) where p, q are positive constants and J : R ∋ x → J(x) ∈ R satisfies (α1) There exist J0 < J1 in (1/2,1) such that J1 > J(x) > J0 for all x ∈ R.

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.

Theorem

. . Assume (α1) holds. Let (x, τ) be a solution of (Ricker) with initial condition (φ, τ0) in X0 ×(0, r] and l : R → R be defined by l(x) = tanh(−1)(1/J(x)−1). Then the following statements are true. (i) If p ≤ µ, then we have lim

t→+∞(x(t), τ(t)) = (0, l(0)).

(ii) If p ∈ (µ, µe], then we have lim

t→+∞(x(t), τ(t)) =

(1 q ln(p/µ), l(1 q ln(p/µ)) ) , where there exists s ∈ [−τ0, 0] so that ϕ(s) > 0; (iii) If p ∈ (µe, µe2], then we have lim

t→+∞(x(t), τ(t)) =

(1 q ln(p/µ), l(1 q ln(p/µ)) ) ;

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.

Theorem

. . (continued) (iv) There exists an infinite sequence {pn}+∞

n=1 of critical values of p in the

interval (µe2, +∞) such that there exist periodic solutions bifurcating from (

1 q ln( pn µ ), l( 1 q ln( pn µ ))

) .

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Example II – The Logistic-Type feedback

Consider      ˙ x(t) = −µx(t)+νx(t −τ(t)) ( 1− x(t −τ(t)) K ) , ˙ τ(t) = 1−J(x(t))(1+tanh(τ(t))), (Logistic) where J : R ∋ x → J(x) ∈ R satisfies (α1).

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For this example, we restrict the initial values in X1 ×(0, r] := (X0 ∩C([−r, 0];[0, K]))×(0, r] where C([−r, 0];[0, K]) is the set of Lipschitz continuous functions ϕ : [−r, 0] → [0, K]. .

Theorem

. . Assume (α1) holds. Let (x, τ) be a solution of (Logistic) with initial condition (φ, τ0) in X1 ×(0, r] and l : R → R be defined by l(x) = tanh(−1)(1/J(x)−1). Then the following statements are true. (i) If ν ∈ (0, µ], then we have lim

t→+∞(x(t), τ(t)) = (0, l(0)).

(ii) If ν ∈ (µ, 2µ], then lim

t→+∞(x(t), τ(t)) = (K(1−µ/ν), l(K(1−µ/ν))),

where there exists s ∈ [−τ0, 0] so that ϕ(s) > 0;

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.

Theorem

. . (continued) (iii) If ν ∈ (2µ, 3µ], then lim

t→+∞(x(t), τ(t)) = (K(1−µ/ν), l(K(1−µ/ν)));

(iv) There exists an infinite sequence {νn}+∞

n=1 of critical values of ν in

(3µ, +∞) such that there exists periodic solutions bifurcated from (K(1−µ/νn), l(K(1−µ/νn))).

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Summary and Remarks

The investigation enables us to . .

1

• btain attractivity of stationary states of the model in question which has

state-dependent delay; . .

2

• bserve Hopf bifurcation when the stationary states lose attractivity;

Future work for state-dependent delays in this direction

• Global Hopf bifurcation problems for the considered models;
• High dimensional problems with various feedback type;
• Plenty of models/equations with state-dependent delay are poorly

understood awaiting our investigation.

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References

• Arino, O., Lhassan, H. M. and Bravo de la P., A mathematical model of

growth of population of fish in the larval stage: density-dependence effects, Math Biosci., 150 (1998), no., 1–20.

• Briat, C., Hjalmarsson, H., et al, Nonlinear state-dependent delay

modeling and stability analysis of internet congestion control, 49th IEEE Conference on Decision and Control, 15-17 Dec. 2010, pp. 1484–1491.

• Hu, Q., Krawcewicz, W. and Turi, J., Stabilization in a state-dependent

model of turning processes, SIAM J. Appl. Math. 72 (2012), 1–24.

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• Driver, R. D., A neutral system with state-dependent delay, J. Differential

Equations, 54 (1984), 73–86.

• Rost, G. and Wu, J., 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 (2007), no. 2086,

2655–2669.

• Bartha, M., Convergence of solutions for an equation with

state-dependent delay, J. Math. Anal. Appl., 254 (2001), 410–432.

• Chen, Y., Global attractivity of a population model with state-dependent

delay, Fields Institute Communications, 36 (2003), 113–118.

• Hu, Q. and Zhao, X.- Q., Global dynamic of a state-dependent model

with unimodal feedback, J. Math. Anal. Appl, 399 (2013) 133-146.

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Thank you for your attention.

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