The Hopf Bifurcation and Its Applications

Marsden, Jerrold E.; McCracken, M.; Sethna, P. R.; Sell, George R.

doi:10.1007/978-1-4612-6374-6

Cited by 1,590 publications

(595 citation statements)

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“…The transition of the dynamical behavior of this delay equation from a stable rest point to a periodic regime as τ increases is easily seen to be the result of a degenerate Hopf bifurcation [35]. Indeed, we recall that x(t) = exp(λ(τ )t) is a solution of the delay equation under consideration if λ is a solution of the characteristic equation λ = −a exp(−λτ ).…”

Section: Delay Equations With Sufficiently Large Delaysmentioning

confidence: 99%

Delay equations and radiation damping

et al. 2001

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Abstract. Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.

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Section: Delay Equations With Sufficiently Large Delaysmentioning

confidence: 99%

Delay equations and radiation damping

et al. 2001

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“…This differential equation describes a nonlinear oscillator. It is well-known (see, e.g., HOLMES & RAND [81] or MARSDEN & MCCRACKEN [116]) that in case the function β is constant and positive, i.e., the system is autonomous, the equilibrium (0, 0) is attractive for α < 0. At α = 0, the system undergoes a Hopf bifurcation: The equilibrium (0, 0) becomes repulsive and an attractive periodic orbit appears.…”

Section: Theorem (Past Hopf Bifurcation)mentioning

confidence: 99%

“…In this section, two-dimensional differential equations which exhibit Hopf bifurcations are studied (see, e.g., MARSDEN & MCCRACKEN [116]). As in the previous section, this bifurcation behavior is transferred to asymptotically autonomous systems.…”

Section: Bifurcations In Dimension Twomentioning

confidence: 99%

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Rasmussen

2007

Lecture Notes in Mathematics

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“…After that, the analysis of Lyapunov values became one of the central problems in considering limit cycles in the neighborhood of equilibrium of two-dimensional dynamical systems (see, e.g. [Marsden & McCracken, 1976;Lloyd, 1988;Yu, 1998;Giné & Santallusia, 2004;Dumortier et al, 2006;Christopher & Li, 2007;Yu & Chen, 2008;Li et al, 2008;Borodzik &Żo ladek, 2008;Yu & Corless, 2009;Li et al, 2012;Giné, 2012;Shafer, 2009] Note that although scholars began to consider the problem of symbolic computation of Lyapunov values (the expressions in terms of coefficients of the right-hand side of the considered dynamical system) in the first half of the last century, substantial progress in the study of Lyapunov values became possible only in the past decade by virtue of the use of modern software tools of symbolic computation. While general expressions for the first and second Lyapunov values (in terms of coefficient expansion of right-hand side of the considered dynamical system) were obtained in the 40-50s of the last century in the works [Bautin, 1949;Serebryakova, 1959], the general expression of the third Lyapunov value was computed only in 2008 [Kuznetsov & Leonov, 2008a; and occupies more than four pages.…”

Section: Lyapunov Values Definitionmentioning

confidence: 99%

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

770

414

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From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

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The Hopf Bifurcation and Its Applications

Cited by 1,590 publications

References 0 publications

Delay equations and radiation damping

Delay equations and radiation damping

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

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