Canonical Melnikov theory for diffeomorphisms

Lomelí, Héctor E.; Meiss, James D.; Ramírez-Ros, Rafael

doi:10.1088/0951-7715/21/3/007

Cited by 12 publications

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“…For a small perturbation to an integrable map with a homoclinic connection to a saddle, however, Melnikov theory can be used to estimate the exit set volume [RKLW90,Wig92]. Melnikov theory, originally developed for flows, was formulated for maps by Easton [Eas84,LMRR08]. Another case that can be treated is that of an adiabatic perturbation, where the lobes cover the region swept by the separatrices of the frozen time subsystems [KW91].…”

Section: A Exit Time Distributionsmentioning

confidence: 99%

Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions. The problem of transport in dynamical systems is to quantify the motion of collections of trajectories between regions of phase space with physical significance, for example, to determine chemical reaction rates, mixing rates in a fluid, or particle confinement times in an accelerator or fusion plasma device. For Hamiltonian systems with a phase space containing regular (elliptic islands and tori) and irregular (chaotic) components, transport is impeded by partial barriers bounding resonance zones or by remnants of invariant tori. In the former case the barriers are formed from the broken separatrices of an unstable periodic orbit, and in the latter, they are formed from similar stable and unstable manifolds of a remnant torus-a cantorus. Thirty years ago, MacKay, Meiss, and Percival discovered that cantori form robust partial barriers for area-preserving maps: in any region of phase space the most resistant barriers are the cantori. In this review, I discuss this work and survey subsequent developments. While much has been done, there are still many open questions.

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Section: A Exit Time Distributionsmentioning

confidence: 99%

Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The first steps towards a discrete Melnikov theory were performed for area-preserving maps [16,17,13,20], and next, for symplectic maps [14], for twist maps [21], for general n-dimension diffeomorphisms [8,24], and for spatial billiard maps [12]. Finally, volume-preserving maps have been considered in [22,23].…”

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“…The Melnikov theory for volume-preserving maps. In this section we shall briefly describe the Melnikov theory for volume-preserving maps developed in [22,23,24].…”

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“…See [22,24] Theorem 1. Under the previous assumptions, there exists a smooth function M : Σ → R, called the Melnikov function, with the following properties.…”

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“…Here, we study the existence of real zeros of the function (24). This function has many properties similar to the ones of elliptic functions and so we shall study its zeros…”

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Separatrix Splitting in 3D Volume-Preserving Maps

Lomelí¹,

Ramírez-Ros²

2008

SIAM J. Appl. Dyn. Syst.

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Abstract. We construct a family of integrable volume-preserving maps in R 3 with a bidimensional heteroclinic connection of spherical shape between two fixed points of saddle-focus type. In other contexts, such structures are called Hill's spherical vortices or spheromaks. We study the splitting of the separatrix under volume-preserving perturbations using a discrete version of the Melnikov method.Firstly, we establish several properties under general perturbations. For instance, we bound the topological complexity of the primary heteroclinic set in terms of the degree of some polynomial perturbations. We also give a sufficient condition for the splitting of the separatrix under some entire perturbations. A broad range of polynomial perturbations verify this sufficient condition. Finally, we describe the shape and bifurcations of the primary heteroclinic set for a specific perturbation.Key words. Separatrix splitting, volume-preserving maps, primary heteroclinic set, Melnikov method, bifurcations AMS subject classifications. 34C37, 34C23, 37C29, 33E201. Introduction. A fundamental question in dynamical systems is the effect that small perturbations of a dynamical system cause on its unperturbed invariant sets. The most studied unperturbed invariant sets are tori and stable/unstable invariant manifolds of hyperbolic sets. Usually, the unperturbed dynamical system is integrable and has separatrices; that is, its stable and unstable invariant manifolds overlap. After a generic perturbation, the perturbed stable and unstable invariant manifolds intersect transversely, which give rise to the onset of chaos, through the creation of Smale horseshoes. This phenomenon is known as the problem of splitting of separatrices. A widely used technique for detecting such intersections is the Melnikov method.Our goal is to apply the Melnikov method to the splitting of separatrices in the discrete volume-preserving framework. Similar questions have been considered before. However, we believe this is the first time that detailed analytical results about the structure of the primary heteroclinic set and its bifurcations are established for specific maps. This represents a step forward with respect to previous works [22,23], in which once written down a formula for the Melnikov function in terms of an infinite series, the approach becomes mainly numerical, because of the technical difficulties that obstruct the analytical one. Here, we have overcome some of these difficulties using basic tools: complex variable theory, quasielliptic functions, homology, and several algebraic tricks. Nevertheless, we have not been able to find an explicit expression of the Melnikov function in terms of elementary functions for any specific perturbation. In opposition, such explicit expressions (in terms of elliptic functions) are known for almost twenty years in the discrete area-preserving setting [17,13].This study is interesting because volume-preserving maps are the simplest, and most natural higher-dimensional versions of the much-studied class o...

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