Expansions in the delay of quasi-periodic solutions for state dependent delay equations

Casal, Alfonso C.; Corsi, Livia; Llave, Rafael de la

doi:10.1088/1751-8121/ab7b9e

Cited by 14 publications

(11 citation statements)

References 40 publications

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“…Formal expansions of periodic and quasiperiodic solutions for small delays were considered in [CCdlL20]. The results of this section establish that the formal expansions of periodic orbits obtained in [CCdlL20] correspond to true periodic orbits and are asymptotic to the true periodic solutions in a very strong sense.…”

Section: The Case Of Small Delaysmentioning

confidence: 73%

“…Remark 4.13. A-posteriori theorems justify asymptotic expansions where solutions are written as formal expansions in terms of the small parameters, see [Chi03,CCdlL20]. Truncations of the formal power series provide approximate solutions.…”

Section: 1mentioning

confidence: 99%

“…Hence, the formal power series in [CCdlL20] are asymptotic in the strong sense that the error in the truncation is bounded by a power of ε, where a stronger norm can be used for smaller ε.…”

Section: The Case Of Small Delaysmentioning

confidence: 99%

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Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

Yang¹,

Gimeno²,

Llave³

2021

Preprint

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We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity.The result also applies to FDEs which are perturbations of some evolutionary partial differential equations(PDEs).The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs.

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Section: The Case Of Small Delaysmentioning

confidence: 73%

Section: 1mentioning

confidence: 99%

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Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

Yang¹,

Gimeno²,

Llave³

2021

Preprint

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“…There are many disciplines such as, but not limited to, population dynamics, ecology, economy, and neural networks [2,3]. Many recent works dedicated to the delay differential equations and the fractional differential equations can be found in [4][5][6][7][8][9][10][11][12][13]. Many numerical studies dedicated to fractional models in epidemiology can be found in [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

et al. 2021

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This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

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“…Recently a number of authors have made substantial progress using the parameterization method to study invariant manifolds in ill posed problems [dlLS19, CdlL20, WdlL20, CGL18], and in particular the method has been used successfully to study periodic and quasi-periodic solutions of SDDEs and their attached stable/unstable manifolds [HdlL16,HdlL17,CCdlL20,YGdlL21,YGdlL]. We think of this as an application of the Poincaré program in problems where the semi-flow theory is underdeveloped or otherwise problematic.…”

Section: Introductionmentioning

confidence: 99%

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

Gimeno¹,

Lessard²,

James³

et al. 2021

Preprint

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A computer-assisted argument is given, which provides existence proofs for periodic orbits in state-dependent delayed perturbations of ordinary differential equations (ODEs). Assuming that the unperturbed ODE has an isolated periodic orbit, we introduce a set of polynomial inequalities whose successful verification leads to the existence of periodic orbits in the perturbed delay equation. We present a general algorithm, which describes a way of computing the coefficients of the polynomials and optimizing their variables so that the polynomial inequalities are satisfied. The algorithm uses the tools of validated numerics together with Chebyshev series expansion to obtain the periodic orbit of the ODE as well as the solution of the variational equations, which are both used to compute rigorously the coefficients of the polynomials. We apply our algorithm to prove the existence of periodic orbits in a state-dependent delayed perturbation of the van der Pol equation.

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Expansions in the delay of quasi-periodic solutions for state dependent delay equations

Cited by 14 publications

References 40 publications

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

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