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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.-…
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The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

TLDR
This review describes mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain.
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Dynamic model

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Structural stability of Lorenz attractors

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://…
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Isochrons and phaseless sets

TLDR
In Winfree's terminology, the existence of isochrons is shown and some of their properties are established and the models leading to these questions concerning dynamical systems are discussed.
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ONE‐DIMENSIONAL DYNAMICS *

At a similar conference sponsored by the New York Academy of Sciences two years ago, we surveyed a range of questions concerning the iteration of quadratic functions.’ These past two years have seen…
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Bifurcation

. The properties of motion close to the transition of a stable family of periodic orbits to complex instability is investigated with two symplectic 4D mappings, natural extensions of the standard…
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Mixed-Mode Oscillations with Multiple Time Scales

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied…
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Singular Hopf Bifurcation in Systems with Two Slow Variables

TLDR
Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable with high amplitude, manifested by a very rapid growth in the amplitude.
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Structurally stable heteroclinic cycles

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields…
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