Stability, Instability and Chaos

Glendinning, Paul

doi:10.1017/cbo9780511626296

Cited by 398 publications

(123 citation statements)

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“…This phenomenon has been well documented for the case when the attractor is a limit cycle (e.g. in the forced Van der Pol equation [42], and in the context of a single FitzHugh system disturbed by noise [43,44,45]). The peculiarity of our situation is that the "canard explosion" happens not to an individual periodic orbit but to the chaotic attractor as a whole (of course, individual unstable periodic orbits embedded into the attractor also experience the explosion, but this remains unnoticed for an observer who watches only the stable sets).…”

Section: From Chaotic To Regular Spikingmentioning

confidence: 81%

Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems

Zaks

Sailer

Schimansky‐Geier

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

105

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We study stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. Diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed in the course of this transition. In order to understand details and mechanisms of noise-induced dynamics we consider a thermodynamic limit N → ∞ of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In the Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.

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Section: From Chaotic To Regular Spikingmentioning

confidence: 81%

Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems

Zaks

Sailer

Schimansky‐Geier

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

105

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“…This results in the 'checkerboard' spatiotemporal pattern shown in Figure 5a. Accordingly, the stress σ(z, t), measured for a given height within the cell (e.g., z = 2/3), displays periodic oscillations with a waveform close to a square wave (and abrupt changes between the low-and high-stress states, as expected for 'relaxation oscillations' [53]). The flipflop period τ flip is of order the structural time τ S .…”

Section: Periodically Oscillating Shear Bandsmentioning

confidence: 85%

Minimal model for chaotic shear banding in shear thickening fluids

Aradian

Cates

2006

Phys. Rev. E

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We present a minimal model for spatiotemporal oscillation and rheochaos in shear-thickening complex fluids at zero Reynolds number. In the model, a tendency towards inhomogeneous flows in the form of shear bands combines with a slow structural dynamics, modelled by delayed stress relaxation. Using Fourier-space numerics, we study the nonequilibrium 'phase diagram' of the fluid as a function of a steady mean (spatially averaged) stress, and of the relaxation time for structural relaxation. We find several distinct regions of periodic behavior (oscillating bands, travelling bands, and more complex oscillations) and also regions of spatiotemporal rheochaos. A low-dimensional truncation of the model retains the important physical features of the full model (including rheochaos) despite the suppression of sharply defined interfaces between shear bands. Our model maps onto the FitzHugh-Nagumo model for neural network dynamics, with an unusual form of long-range coupling.

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“…In the one-sided case, the orbit apporaches the bifurcation point from one side of the bifurcation point as its period tends to infinity. In the two-sided case, such as the Shil'nikov case [1], the locus of the orbit in parameter space oscillates about the bifurcation value creating the so-called 'Shil'nikov wiggle' as the period of the orbit tends to infinity. Moreover, there are period-doubling and reverse period-doubling bifurcations of the orbit together with more complicated homoclinic bifucations.…”

Section: Introductionmentioning

confidence: 99%