Multimode oscillations in a modified Van Der Pol oscillator containing a positive nonlinear conductance

Shinriki, Masanori; Yamamoto, Masaomi; Mori, Shohei

doi:10.1109/proc.1981.11973

Cited by 94 publications

(26 citation statements)

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“…The first one is the Shinriki oscillator [8,9]. The dynamics of the second oscillator [10] is time-delay induced and mimics the dynamics of the Mackey-Glass equation.…”

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confidence: 99%

Tool to recover scalar time-delay systems from experimental time series

et al. 1996

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We propose a method that is able to analyze chaotic time series, gained from experimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form= h(y(t), y(t−τ0)), the delay time τ0 and the function h can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators. P.A.C.S.: 05.45.+bTime series analysis of chaotic systems has gained much interest in recent years. Especially, embedding of time series in a reconstructed phase space with the help of time-delayed coordinates was widely used to estimate fractal dimensions of chaotic attractors [1, 2] and Lyapunov exponents [3]. It is the advantage of embedding techniques that the time series of only one variable has to be analyzed, even if the investigated system is multi-dimensional. Furthermore, it can be applied, in principle, to any dynamical system. Unfortunately, the embedding techniques only yield information, if the dimensionality of the chaotic attractor under investigation is low. Another drawback is the fact that it does not give any information about the structure of the dynamical system, in the sense, that one is able to identify the underlying instabilities. In the following, we propose a method which is taylor-suited to identify scalar systems with a time-delay induced instability. We will show that the differential equation can be recovered from the time series, if the investigated dynamics obeys a scalar time-delay differential equation. There are no restrictions to the dimensionality of the chaotic attractor. Additionally, the method has the advantage to be insensitive to noise.We consider the time evolution of scalar time-delay differential equationsẏ (t) = h(y(t), y(t − τ 0 )), (1) * published in Phys. Rev. E 54 (1996) R3082.

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“…The first one is the Shinriki oscillator [8,9]. The dynamics of the second oscillator [10] is time-delay induced and mimics the dynamics of the Mackey-Glass equation.…”

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confidence: 99%

Tool to recover scalar time-delay systems from experimental time series

et al. 1996

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“…Its dependence on the delay time t has to be of resonance type, with the minima at t coincid ing with the periods 7] of the distinct periodic orbits. We have used an electronic autonomous chaos os cillator suggested by Shinriki et al [21], in order to test the method at a real experimental situation. The circuit is shown in Fig.…”

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confidence: 99%

Delayed Feedback Control of Chaos in an Electronic Double-Scroll Oscillator

Kittel

Parisi

Pyragas

et al. 1994

Zeitschrift Für Naturforschung A

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We present experimental results on stabilizing unstable periodic orbits of an autonomous chaos oscillator based on a simple electronic circuit. Control is achieved by applying the difference between the actual and a delayed output signal of the oscillator. The quality of chaos control can be measured via the strength of perturbation. The dependence on the delay time shows a characteristic resonancetype behavior.Since the pioneer work done by Hübler and coworkers [1], controlling chaos has become a fasci nating topic in the field of nonlinear dynamics. In particular, the method introduced by Ott, Grebogi, and Yorke (OGY) [2] develops to an important tool to overcome undesired chaotic behavior in various fields of applications. The key observation is that the typical chaotic attractor embeds an infinite number of un stable periodic orbits of distinct periods Tt. They can be stabilized by an only small, carefully chosen, feed back perturbation applied to some system parameter available for external adjustment. The OGY approach is as follows; At first, one determines some of the unstable low-periodic orbits of the strange attractor by finding the fixed points in the Poincare map of the system. Then the linear transformations close to these points and their dependence on the parameter varia tion have to be reconstructed from experiment. Fi nally, a small time-dependent perturbation is applied to the system, in order to stabilize these already exist ing periodic orbits.The O GY method is a very general one. It does not require any a priori analytical knowledge of the sys tem dynamics and has been successfully applied to various physical experiments including a magnetic ribbon [3], a spin-wave system [4], a chemical system [5], an electric diode [6], laser systems [7], and cardiac systems [8].

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“…In the context of chaotic circuits, such topology was originally proposed in [SYM81], and was studied afterwards in [FGA84] and [FRGP93] in the case R 0 = 0 and assuming a nonlinear positive conductance for the resistor R. With slight modifications, this circuit has been extensively studied in the last two decades; see [GK92] or [HBCJM91]. Taking R 0 = 0 and substituting the nonlinear element by the so-called Chua diode, many papers have also been written; see [CWHZ93], [Ma93], and references therein.…”

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confidence: 99%

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

Freire¹

2005

SIAM J. Appl. Math.

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Abstract. The birth of limit cycles in 3D (three-dimensional) piecewise linear systems for the relevant case of symmetrical oscillators is considered. A technique already used by the authors in planar systems is extended to cope with 3D systems, where a greater complexity is involved.Under some given nondegeneracy conditions, the corresponding theorem characterizing the bifurcation is stated. In terms of the deviation from the critical value of the bifurcation parameter, expressions in the form of power series for the period, amplitude, and the characteristic multipliers of the bifurcating limit cycle are also obtained.The results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3D electronic circuit genealogically related to the classical Van der Pol oscillator. Key words. piecewise linear systems, bifurcation theory, limit cycles AMS subject classifications. 37G15, 34C15 DOI. 10.1137/0406061071. Introduction and main results. Piecewise linear modeling of nonlinear dynamical systems is especially successful in some engineering problems, such as the analysis and design of electronic oscillators or control systems (see, e.g., [CFPT02]). However, in the framework of piecewise linear systems, there are no general bifurcation results explaining the appearance or disappearance of self-sustained oscillations, as is the case for the Hopf bifurcation theorem in the context of differentiable systems. Thus, the authors gave in [FPR99] a complete characterization of the focus-centerlimit cycle bifurcation for symmetric planar piecewise linear systems. Now we show how the corresponding result can be extended to the 3D case.We consider a common situation in applications, namely, dynamical systems defined by piecewise continuous vector fields with three linear zones and two parallel frontiers. Furthermore, it is assumed that such systems show symmetry with respect to the origin; that is, if we put them in the form dx/dτ = f (x) with x ∈ R 3 , they satisfy f (−x) = −f (x). In particular, f (0) = 0, and so the origin is an equilibrium point for all values of the parameters. By means of a linear change of variables, it is always possible to suppose that the frontiers are the planes Σ 1 = {x ∈ R 3 : x 1 = 1} and Σ −1 = {x ∈ R 3 : x 1 = −1}. We denote by L (left), C (central), and R (right) the regions of R 3 at which x 1 < −1, |x 1 | ≤ 1, and x 1 > 1, respectively, hold. To be more precise, we consider systems expressed as follows:

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Multimode oscillations in a modified Van Der Pol oscillator containing a positive nonlinear conductance

Cited by 94 publications

References 3 publications

Tool to recover scalar time-delay systems from experimental time series

Tool to recover scalar time-delay systems from experimental time series

Delayed Feedback Control of Chaos in an Electronic Double-Scroll Oscillator

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

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