Hopf Bifurcation and Stability of Periodic Solutions of Differential-difference and Integro-differential Equations

Kazarinoff, Nicholas D.; Wan, Yieh-Hei; Driessche, P. van den

doi:10.1093/imamat/21.4.461

Cited by 55 publications

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“…Whereas Nussbaum's approach uses the transition from a stable to an ejective fixed point (with respect to a certain set), Hassard and Wang [7,13] can explain this bifurcation as a Hopf bifurcation. After Nussbaum's result had appeared, H.O.…”

Section: Yc (T) = -C~f (X (T -1))mentioning

confidence: 99%

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Hadeler

1980

Numer. Math.

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Section: Yc (T) = -C~f (X (T -1))mentioning

confidence: 99%

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Hadeler

1980

Numer. Math.

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“…The Banach space setting of continuous functions has been extensively used in the study of bifurcations arising from DDEs, see e.g. [8,17,19,21,37,47,60], while the Hilbert space setting is typically adopted for the approximation of DDEs or their optimal control; see e.g. [1-3, 9, 22, 28, 33, 35, 36, 42].…”

Section: Recasting a Dde Into An Infinite-dimensional Evolution Equatmentioning

confidence: 99%

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Chekroun¹,

Kröner²,

Liu³

2018

Hamilton-Jacobi-Bellman Equations

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Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation. 26 References 26

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“…Many of them [8][9][10] have done systematic research to the price differential equation model in the market economy and have got many essential results [10].In 1997, the literature [1] studied the price reyleigh equation model without considering the effect of finite delay. …”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation In Numerical Approximation For Price Reyleigh Equation With Finite Delay

Taheri-Dehkordi¹,

Aliasghari-Namini²

2014

J. Math. Computer Sci.

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The numerical approximation of Price Reyleigh equation is considered using delay as parameter. Fist, the delay difference equation obtained by using Euler method is written as a map.According to the theories of bifurcation for discrete dynamical systems, the conditions to guarantee the existence of Hopf bifurcation for numerical approximation are given. The relations of Hopf bifurcation between the continuous and the discrete are discussed. That when the Price Reyleigh equation has Hopf bifurcations at , the numerical approximation also has Hopf bifurcations at is proved.

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Hopf Bifurcation and Stability of Periodic Solutions of Differential-difference and Integro-differential Equations

Cited by 55 publications

References 0 publications

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Effective computation of periodic orbits and bifurcation diagrams in delay equations

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Hopf Bifurcation In Numerical Approximation For Price Reyleigh Equation With Finite Delay

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