The inner equation for generic analytic unfoldings of the Hopf-zero singularity

Baldomá, Inmaculada; Seara, Tere M.

doi:10.3934/dcdsb.2008.10.323

Cited by 14 publications

(13 citation statements)

References 16 publications

(22 reference statements)

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“…This bifurcation is known to give rise to a Hopf bifurcation [39]. It is a singular Hopf bifurcation [9,26] due to the fact that the linearization (upon blowup) has eigenvalues of the form ∼ ±iω, ∼ λ, ω, λ = 0 as → 0 at the Hopf bifurcation; it is therefore a zero-Hopf bifurcation [2,3] for = 0.…”

Section: Introductionmentioning

confidence: 99%

The Dud Canard: Existence of Strong Canard Cycles in R3

Kristiansen

2023

Preprint

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Section: Introductionmentioning

confidence: 99%

The Dud Canard: Existence of Strong Canard Cycles in R3

Kristiansen

2023

Preprint

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“…Moreover, we provide an asymptotic formula for the difference between these two solutions of the inner equation. We follow the approach presented in [BS08].…”

Section: Strategy For the Proof Of Theorem Amentioning

confidence: 99%

“…Due to the slow-fast character of the system, to capture the asymptotic first order of the difference ∆z = z u − z s , we need to give the main terms of this difference close to the singularities, concretely, up to distance of order δ 2 . To this end, we derive the inner equation, see [Bal06;BS08], which contains the first order of the Hamiltonian H sep (see (I.2.30)) close to (one of) the singularities and is independent of the small parameter δ. That is, we look, for instance, for a Hamiltonian which is a good approximations of H sep in a neighborhood of u = iA.…”

Section: I23 Derivation Of the Inner Equationmentioning

confidence: 99%

“…In Section I.5.2, we provide the asymptotic formula for the difference ∆Z 0 = Z u 0 − Z s 0 given in (I.2.45). For both parts, we follow the approach given in [BS08].…”

Section: Analysis Of the Inner Equationmentioning

confidence: 99%

“…We provide suitable solutions Z u,s 0 (U ) of the so-called inner equation. The inner equation, see [Bal06;BS08], describes the dominant behavior of the functions z u and z s close to (one of) the singularities u = ±iA. In particular, it involves the first order of the Hamiltonian H sep close to a singularity and it is independent of the small parameter δ.…”

Section: Ii33 a First Order Of The Invariant Manifolds Near The Singu...mentioning

confidence: 99%

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Homoclinic and chaotic phenomena to L3 in the restricted 3-Body Problem

Giralt Miron

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(English) The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom autonomous Hamiltonian system with five critical points, L1,…...,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest one. This thesis focuses on the study of the one dimensional unstable and stable manifolds associated to L3 and the analysis of different homoclinic and chaotic phenomena surrounding them. We assume that the ratio between the masses of the primaries is small. First, we obtain an asymptotic formula for the distance between the unstable and stable manifolds of L3. When the ratio between the masses of the primaries is small the eigenvalues associated with L3 have different scales, with the modulus of the hyperbolic eigenvalues smaller than the elliptic ones. Due to this rapidly rotating dynamics, the invariant manifolds of L3 are exponentially close to each other with respect to the mass ratio and, therefore, the classical perturbative techniques (i.e. the Poincaré-Melnikov method) cannot be applied. In fact, the formula for the distance between the unstable and stable manifolds of L3 relies on a Stokes constant which is given by the inner equation. This constant can not be computed analytically but numerical evidences show that is different from zero. Then, one infers that there do not exist 1-round homoclinic orbits, i.e. homoclinic connections that approach the critical point only once. The second result of the thesis concerns the existence of 2-round homoclinic orbits to L3, i.e. connections that approach the critical point twice. More concretely, we prove that there exist 2-round connections for a specific sequence of values of the mass ratio parameters. We also obtain an asymptotic expression for this sequence. In addition, we prove that there exists a set of Lyapunov periodic orbits whose two dimensional unstable and stable manifolds intersect transversally. The family of Lyapunov periodic orbits of L3 has Hamiltonian energy level exponentially close to that of the critical point L3. Then, by the Smale-Birkhoff homoclinic theorem, this implies the existence of chaotic motions (Smale horseshoe) in a neighborhood exponentially close to L3 and its invariant manifolds. In addition, we also prove the existence of a generic unfolding of a quadratic homoclinic tangency between the unstable and stable manifolds of a specific Lyapunov periodic orbit, also with Hamiltonian energy level exponentially close to that of L3. (Català) El problema restringit dels 3 cossos modela el moviment d'un cos de massa negligible que es troba sota la influència gravitatòria de dos cossos massius anomenats primaris. Si els primaris realitzen moviments circulars i el cos sense massa és coplanar amb ells, es té el problema restringit planar i circular dels 3 cossos (RPC3BP). En coordenades sinòdiques, aquest és un sistema Hamiltonià autònom de dos graus de llibertat i té cinc punts crítics, L1,..,L5, anomenats punts de Lagrange. El punt de Lagrange L3 és un punt crític de tipus centre-sella, col·lineal amb els primaris i que es troba al cantó oposat del primari petit respecte del gran. Aquesta tesi estudia les varietats unidimensionals inestable i estable associades a L3 i analitza alguns dels diferents fenòmens homoclínics i caòtics que les envolten. A més, suposarem que la ràtio entre les masses dels primaris és petita. Primerament, obtenim una fórmula asimptòtica per a la distància entre les varietats inestable i estable de L3. Quan la ràtio entre les masses dels primaris és petita, els valors propis associats a L3 tenen escales diferents; és a dir, el mòdul dels valors propis hiperbòlics és més petit que el dels el·líptics. Degut a aquesta dinàmica de rotació ràpida, les varietats invariants de L3 es troben exponencialment properes l'una de l'altre respecte a la ràtio de masses i, per tant, les tècniques pertorbatives clàssiques (és a dir, el mètode de Poincaré-Melnikov) no apliquen. És més, la fórmula per a la distància entre les varietats inestable i estable de L3 ve donada per una constant de Stokes obtinguda mitjançant l'anomenada equació inner. Aquesta constant no es pot calcular analíticament, tot i així, evidències numèriques mostren que és diferent de zero. D'aquest resultat és pot inferir que no existeixen òrbites homoclíniques de 1 volta, és a dir, connexions homoclíniques que s'apropen al punt crític només una vegada. El segon resultat de la tesi estudia l'existència d'òrbites homoclíniques a L3 de 2 voltes, és a dir, connexions que s'acosten dues vegades al punt crític. Més concretament, demostrem que existeixen connexions de 2 voltes per a una successió específica de valors de la ràtio de masses tendint a zero i obtenim una expressió asimptòtica per a aquesta successió. Endemés, demostrem que existeix un conjunt d'òrbites periòdiques de Lyapunov les varietats inestables i estables bidimensionals de les quals es tallen transversalment. Aquest conjunt d'òrbites periòdiques de Lyapunov de L3 té un nivell d'energia Hamiltonià exponencialment proper al del punt crític L3. Per tant, segons el teorema homoclínic de Smale-Birkhoff, això implica l'existència de moviments caòtics (és a dir, d'una ferradura de Smale) en un entorn exponencialment proper de L3 i les seves varietats invariants. A més, també demostrem l'existència del desplegament genèric d'una tangència quadràtica homoclínica entre les varietats inestable i estable associades a una òrbita periòdica de Lyapunov concreta, també amb un nivell d'energia Hamiltonià exponencialment proper al de L3.

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Zero-Hopf bifurcation in the generalized Michelson system

Llibre

Makhlouf

2016

Chaos, Solitons & Fractals

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The inner equation for generic analytic unfoldings of the Hopf-zero singularity

Cited by 14 publications

References 16 publications

The Dud Canard: Existence of Strong Canard Cycles in R3

The Dud Canard: Existence of Strong Canard Cycles in R3

Homoclinic and chaotic phenomena to L3 in the restricted 3-Body Problem

Zero-Hopf bifurcation in the generalized Michelson system

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