Canards Existence in the Hindmarsh–Rose model

Ginoux, Jean-Marc; Llibre, Jaume; Tchizawa, Kiyoyuki

doi:10.1051/mmnp/2019012

Cited by 8 publications

(4 citation statements)

References 31 publications

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“…Hence the normalized slow dynamics has a pseudo singular point of saddle-type. It follows from Proposition 3.4 in [21] that the singularly perturbed system admit a canard solution.…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 96%

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Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag τ between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power h and the relaxation of the instability δ do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient η just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

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“…Hence the normalized slow dynamics has a pseudo singular point of saddle-type. It follows from Proposition 3.4 in [21] that the singularly perturbed system admit a canard solution.…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 96%

“…where the dot is the derivation with respect to t ∈ C. We make the time scale transformationt = e −ηt to transform (21) into an equation with rational coefficients.…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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“…In recent publications, a new approach to n-dimensional singularly perturbed systems of ordinary differential equations, called the Flow Curvature Method, has been developed by [34][35][36][37]. It consists of considering the trajectory curves integral of such systems as curves in Euclidean n-space.…”

Section: Slow Invariant Manifoldmentioning

confidence: 99%

Slow Invariant Manifold of Laser with Feedback

Ginoux

Meucci

2021

Symmetry

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Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime.

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“…In (Corson and Aziz-Alaoui, 2009) they proceed the local stability and the numerical asymptotic analysis of Hindmarsh-Rose model are then developed in order to comprehend bifurcations and dynamics evolution of a single Hindmarsh-Rose neuron. The authors in (Ginoux et al, 2019) work for the extend method which improves the classical ones used to the case of three-dimensional singularly perturbed systems with two fast variables. This method state a unique generic condition for the existence of canard solutions for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a wellknown property of linear algebra.…”

Section: Introductionmentioning

confidence: 99%

Existence Canard Solutions For Four Dimensional Hindmarsh-Rose Model with Respect to Infinitesimal Parameter

2024

ZJPAS

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This research endeavor seeks to investigate the presence and viability of canard solutions within the context of the generalized Hindmarsh-Rose model, specifically when extended to four dimensions. To achieve this, nonstandard analysis is employed as a powerful tool for identifying and characterizing canard solutions within the four-dimensional singularly perturbed system, wherein two fast variables are considered in the folded saddle case. By undertaking this rigorous approach, we aim to contribute valuable insights to the understanding of canard phenomena in complex dynamical systems.

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Canards Existence in the Hindmarsh–Rose model

Cited by 8 publications

References 31 publications

Complex dynamics in a quasi-periodic plasma perturbations model

Complex dynamics in a quasi-periodic plasma perturbations model

Slow Invariant Manifold of Laser with Feedback

Existence Canard Solutions For Four Dimensional Hindmarsh-Rose Model with Respect to Infinitesimal Parameter

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