Global topological properties of the Hopf bifurcation

Physica D: Nonlinear Phenomena

2020

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In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the system and the role of the strange set in these decompositions. We calculate the corresponding Morse equations and study their change along the successive bifurcations. In addition, we formulate and prove some theorems which are applicable in more general situations. These theorems refer to Poincaré-Andronov-Hopf bifurcations of arbitrary codimension, bifurcations with two homoclinic loops and a study of the role of the travelling repellers in the transformation of repeller-attractor pairs into attractor-repeller ones.

“…Our main references for the Conley index theory are [9,11,49,45]. Some applications of the Conley index theory to the study of the Lorenz equations can be seen in [52,53,21].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Generalized Pitchfork Bifurcationsmentioning

confidence: 99%

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A Conley index study of the evolution of the Lorenz strange set

Physica D: Nonlinear Phenomena

2020

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“…. For a discussion of generalized Poincare -Andronov -Hopf bifurcations we refer the reader to the paper [41] .…”

Section: Dynamics Of Plane Continuamentioning

confidence: 99%

Unstable manifold, Conley index and fixed points of flows

Journal of Mathematical Analysis and Applications

2014

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We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.

“…We prove that, for flows on compact connected and oriented differentiable manifolds with trivial 1-dimensional cohomology, dynamical robustness is equivalent to topological robustness and, as a consequence, the preservation of non-saddleness can be detected by topological means. It can be seen that non-saddle sets are involved in generalized Andronov-Poincaré-Hopf bifurcations (in the sense of [37]) but this will be the subject of a different paper. Our point of view is mainly topological and has deep connections with the Conley index theory.…”

Section: Introductionmentioning

confidence: 99%

Dissonant points and the region of influence of non-saddle sets

Journal of Differential Equations

2020

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The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set. We see, in particular, that some topological conditions are sufficient to guarantee that these sets are attractors or repellers. We study in detail the existence of dissonant points of the flow, which play a key role in the description of the region of influence of a non-saddle set. These points are responsible for much of the dynamical and topological complexity of the system. We also study non-saddle sets from the point of view of the Conley index theory and consider, among other things, the case of flows on manifolds with trivial first cohomology group. For flows on these manifolds, dynamical robustness is equivalent to topological robustness. We carry out a particular study of 2-dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces. We also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers. MSC: 37C75, 37B30.Non-saddle set, region of influence, dissonant point, Conley index, homoclinic orbit, robustness.