A modulus of 3-dimensional vector fields

Togawa, Yoshio

doi:10.1017/s0143385700004028

Cited by 7 publications

(2 citation statements)

References 9 publications

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“…The saddle quantity, whether larger than one or not, is always a topological invariant of saddle-focus homoclinic orbits [21,67,116,395]. The proof of Togawa [395] uses link types of period orbits for saddle quantities smaller than one. Dufraine [116] proved that the absolute value of Im ν s is also a modulus.…”

Section: Hypothesis 51 (Eigenvalue Conditions) Consider the Followimentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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Section: Hypothesis 51 (Eigenvalue Conditions) Consider the Followimentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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“…Regarding the construction of invariants under conjugacy for homoclinic cycles of a vector field, we refer the reader to [21], where Togawa analyzes a homoclinic cycle of a saddle-focus and shows, using a knot-like argument, that the saddle-index is a conjugacy invariant; to the paper [1], where Arnold et al prove that the saddle-index is in fact an invariant under topological equivalence; and to the work [7] whose author, in the same setting, describes a new invariant under conjugacy given by the absolute value of the imaginary part of the complex eigenvalues of the saddle-focus. The search for a complete set of invariants for more general homoclinic cycles associated to either a saddle-focus or a periodic solution is still an open problem.…”

Section: Final Remarkmentioning

confidence: 99%

Moduli of stability for heteroclinic cycles of periodic solutions

Carvalho¹,

Lohse²,

Rodrigues³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider C 2 vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

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