Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation

Sekikawa, Munehisa; Inaba, Nobuyuki; Tsubouchi, Takashi

doi:10.1016/j.physd.2004.02.012

Cited by 24 publications

(25 citation statements)

References 15 publications

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“…This occurs when the units force each other close to the Andronov-Hopf bifurcation where the divergence of the phase flow is high. 25,26 This phenomenon is related to the existence of an area of canard explosion. 27,28 Let us summarize our findings concerning the behavior of the two-oscillator system: ͑i͒ sharing the common power supply via a resistor, identical coupled circuits tend to attain antiphase synchronization; ͑ii͒ the antiphase locking pattern may maintain self-sustained dynamics even outside of the corresponding parameter range for the individual system; and ͑iii͒ this effect is preserved in the presence of a mismatch of reasonable size between the coupled units.…”

Section: -3mentioning

confidence: 99%

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Multimode dynamics in a network with resource mediated coupling

Postnov

Sosnovtseva

Scherbakov

et al. 2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The purpose of this paper is to study the special forms of multimode dynamics that one can observe in systems with resource-mediated coupling, i.e., systems of self-sustained oscillators in which the coupling takes place via the distribution of primary resources that controls the oscillatory state of the individual unit. With this coupling, a spatially inhomogenous state with mixed high and lowamplitude oscillations in the individual units can arise. To examine generic phenomena associated with this type of interaction we consider a chain of resistively coupled electronic oscillators connected to a common power supply. The two-oscillator system displays antiphase synchronization, and it is interesting to note that two-mode oscillations continue to exist outside of the parameter range in which oscillations occur for the individual unit. At low coupling strengths, the multioscillator system shows high dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths, one typically observes spatial clustering involving a few oscillating units. We describe three different scenarios according to which the cluster can slide along the chain as the bias voltage changes. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2805194͔ Systems of interacting nonlinear oscillators play an important role in many areas of science and technology. A rat kidney, for instance, contains of the order of 30 000 nephrons, each of which tends to display large amplitude oscillations in the regulation of its incoming blood flow. The arteriolar blood pressure controls the oscillatory state of the individual nephron. At the same time, the nephrons interact via the displacement of blood from one nephron to its neighbors produced by this oscillatory regulation. We refer to such structures, in which the oscillatory dynamics of the individual unit can no longer be separated from the coupling among the units, as a system with resource-mediated coupling. To examine some of the generic aspects of this type of system, the paper considers a chain of resistively coupled electronic oscillators sharing a common power supply. For low coupling strengths, the system tends to produce multimode dynamics in the form of high-dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths the resource distributed chain displays spatial clustering involving a small number of oscillatory units. In all cases the distribution of individual amplitudes is highly inhomogeneous, showing an interesting example of spatiotemporal mixed-mode oscillations. We study the mechanisms by which this type of cluster slides along the chain as the external bias voltage is changed.

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Section: -3mentioning

confidence: 99%

“…When forced by external regular or noisy signals, a system in the canard region is known to produce chaos. 25,26,29 Hence, the scenario A for changes of cluster location includes two regions of chaotic behavior related to the extinction and to the onset of oscillations in edge units.…”

Section: Change Of Cluster Locationmentioning

confidence: 99%

Multimode dynamics in a network with resource mediated coupling

Postnov

Sosnovtseva

Scherbakov

et al. 2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We know that their detailed analysis is extremely important not only from the practical viewpoint but also from the physical viewpoint. There are many studies reporting the nonlinear phenomena in the IDS from the physical viewpoint [7][8][9][10][11][12][13]. We have also proposed a simple model, which simulates the dynamic behavior of the current mode controlled DC/DC converter, and have essentially investigated the bifurcation phenomena including the border-collision bifurcation [14,15].…”

Section: Introductionmentioning

confidence: 99%

The Stabilizing mechanism for an interrupted dynamical system with periodic threshold

Asahara

Tasaki

Aihara

et al. 2012

NOLTA

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a) asahara@bifurcation.jp b) tasaki@bifurcation.jp c) aihara@sat.t.u-tokyo.ac.jp d ) takuji@oita-u.ac.jpAbstract: This study mathematically analyzes an interrupted dynamical system (IDS) with a periodic threshold. First, we describe a simple IDS, which is dependent on its own state and a periodic interval, and explain the behavior of the waveform. Then, we define the discrete map (return map) of the system and calculate the bifurcation diagrams. Finally, we focus on the dynamical structure of the return map in the system with a periodic threshold and discuss the stabilizing mechanism, especially its effect in a wide parameter space. The stabilizing effect is verified by the laboratory experiment.

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“…This problem seems to be an attractive hot topic [3][4][5][6] since the solution is extremely sensitive to the fluctuation of B 0 .…”

Section: Introductionmentioning

confidence: 99%

“…The canard, being investigated in various fields of science [1][2][3][4][5][6][7][8][9][10][11][12], was originally discovered by Benoit et al [1]. It is clarified that the amplitude of the limit cycle in Ref.…”

Section: Introductionmentioning

confidence: 99%