Cascades of reversible homoclinic orbits to a saddle-focus equilibrium

Härterich, Jörg

doi:10.1016/s0167-2789(97)00210-8

Cited by 43 publications

(37 citation statements)

References 7 publications

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“…Therefore either for λ > 0 or for λ < 0 there exists a unique 1-homoclinic orbit to p 2 . We remark that in either case the results in [7] imply that a complicated set of N -homoclinic orbits exists whenever a 1-homoclinic to the saddle-focus p 2 exists.…”

Section: More About the Dynamics Near The Heteroclinic Cycle γmentioning

confidence: 85%

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Homoclinic snaking near a heteroclinic cycle in reversible systems

Knobloch

Wagenknecht

2005

Physica D: Nonlinear Phenomena

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Snaking curves of homoclinic orbits have been found numerically in a number of ODE models from water wave theory and structural mechanics. Along such a curve infinitely many fold bifurcation of homoclinic orbits occur. Thereby the corresponding solutions spread out and develop more and more bumps (oscillations) about their own centre. A common feature of the examples is that the systems under consideration are reversible.In this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria, one of which is a bi-focus. Using Lin's method a snaking of 1-homoclinic orbits is proved to occur in an unfolding of such a cycle. Further dynamical consequences are discussed.As an application a system of Boussinesq equations is considered, where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi-focus equilibrium.

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Section: More About the Dynamics Near The Heteroclinic Cycle γmentioning

confidence: 85%

“…In this case each 1-homoclinic orbit to p 1 is accompanied by infinitely many N -homoclinic orbits for each N ∈ N, see [7]. Again, all these orbits form homoclinic bellows such that an abundance of complexity can be found near the cycle.…”

Section: More About the Dynamics Near The Heteroclinic Cycle γmentioning

confidence: 96%

Homoclinic snaking near a heteroclinic cycle in reversible systems

Knobloch

Wagenknecht

2005

Physica D: Nonlinear Phenomena

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“…For instance, a solitary wave exist for ϕ ≈ 100.3 • and its structure is given in panels (b)-(d). Interestingly, it can be proved that the existence of one solitary wave for a given value of the physical parameters implies the existence of infinitely many others if the system is reversible and the upstream state is a focus-focus [28]. Such a theoretical result, which was demonstrated earlier for conservative systems [29], is a consequence of the spiralling linear dynamics due to the complex eigenvalue and the additional orbits are like…”

Section: B Saddle-saddle and Focus-focus Domainsmentioning

confidence: 75%

Structure and evolution of magnetohydrodynamic solitary waves with Hall and finite Larmor radius effects

et al. 2019

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Nonlinear and low-frequency solitary waves are investigated in the framework of the onedimensional Hall-magnetohydrodynamic model with finite Larmor effects and a double adiabatic model for plasma pressures. The organization of these localized structures in terms of the propagation angle with respect to the ambient magnetic field θ and the propagation velocity C is discussed. There are three types of regions in the θ −C plane that correspond to domains where either solitary waves cannot exist, are organized in branches, or have a continuous spectrum. A numerical method valid for the two latter cases, that rigorously proves the existence of the waves, is presented and used to locate many waves, including bright and dark structures. Some of them belong to parametric domains where solitary waves were not found in previous works. The stability of the structures has been investigated by first performing a linear analysis of the background plasma state and second by means of numerical simulations. They show that the cores of some waves can be robust but, for the parameters considered in the analysis, the tails are unstable. The substitution of the double adiabatic model by evolution equations for the plasma pressures appears to suppress the instability in some cases and to allow the propagation of the solitary waves during long times.Exact solitary waves solutions in the Hall-MHD model for cold [21] and warm plasmas with scalar [22] and double-adiabatic pressure models [12] have been also found.In the case of the Hall-MHD model with a double adiabatic pressure tensor, the traveling wave ansatz leads to a pair of coupled ordinary differential equations that governs the normalized components of the magnetic field normal to the propagation direction, named b y and b z . Such a system has a hamiltonian structure and is reversible, i.e. solutions are invariant under the transformation (ζ, b y , b z → −ζ, −b y , b z ), with ζ the independent variable. Adding FLR effects does not change the reversible character of the dynamical system but it increases the effective dimension from two to four [23]. Numerical evidence about the existence of solitary waves in the parametric domain where the upstream state is a saddle-center was also given [23]. The hamiltonian character of the dynamical system with FLR effects is an open and interesting topic, especially because an energy conservation theorem is not known for the Hall-MHD model with double adiabatic pressure and without FLR effects.This work investigates the existence and stability of solitary waves in the FLR-Hall-MHD model with double adiabatic pressure tensor. Section II follows Ref.[23] closely, and presents in a concise way the procedure to find the dynamical system that governs the solitary waves. The details of the method are given in Appendix A, where few discrepancies with the results of Ref.[23] are highlighted. Section II also discusses the main properties of the dynamical system and takes advantage of some geometrical arguments related with the dimension, reversible ...

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“…Equation (3.1) is invariant under the transformation z → −z and thus it is a reversible system. In this section, we use the theory of reversible systems [16,19,23,24,13,14] to characterize the homoclinic orbits to the fixed point of (3.1), which correspond to pulses or solitary waves of the Ostrovsky equation in various regions of the (p, q) plane.…”

Section: Solitary Waves and Local Bifurcationsmentioning

confidence: 99%

Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation

Choudhury

Ivanov

Liu

2007

Chaos, Solitons & Fractals

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The Ostrovsky equation is a model for gravity waves propagating down a channel under the influence of Coriolis force. This equation is a modification of the famous Korteweg-de Vries equation and is also Hamiltonian. However the Ostrovsky equation is not integrable and in this contribution we prove its nonintegrability. We also study local bifurcations of its solitary waves.MSC: 35Q35, 35Q53, 37K10

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Cascades of reversible homoclinic orbits to a saddle-focus equilibrium

Cited by 43 publications

References 7 publications

Homoclinic snaking near a heteroclinic cycle in reversible systems

Homoclinic snaking near a heteroclinic cycle in reversible systems

Structure and evolution of magnetohydrodynamic solitary waves with Hall and finite Larmor radius effects

Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation

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