DIFFUSION, INTERMITTENCY, AND NOISE-SUSTAINED METASTABLE CHAOS IN THE LORENZ EQUATIONS: EFFECTS OF NOISE ON MULTISTABILITY

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

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Chaotic Oscillation via Edge of Chaos Criteria

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741730035x ◽

2017 ◽

Vol 27 (11) ◽

pp. 1730035 ◽

Cited By ~ 2

Author(s):

Makoto Itoh ◽

Leon Chua

Keyword(s):

Dynamical Systems ◽

Stable Equilibrium ◽

Nonlinear Dynamical Systems ◽

Chaotic Oscillation ◽

Stable Equilibrium Point ◽

Chaotic Oscillations ◽

Periodic Forcing ◽

Nonlinear Dynamical ◽

Edge Of Chaos ◽

Forced Oscillators

In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh–Nagumo model, the Hodgkin–Huxley model, and the Morris–Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit [Formula: see text] connected across a current source.

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Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414300201 ◽

2014 ◽

Vol 24 (07) ◽

pp. 1430020 ◽

Cited By ~ 2

Author(s):

Paulo C. Rech

Keyword(s):

Dynamical Systems ◽

Continuous Time ◽

Nonlinear Dynamical Systems ◽

Angular Frequency ◽

Two Dimensional ◽

Periodic Forcing ◽

Dimensional Parameter ◽

Parameter Spaces ◽

Nonlinear Dynamical ◽

First Order

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

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Hopf Bifurcations in Complex Multiagent Activity: The Signature of Discrete to Rhythmic Behavioral Transitions

Brain Sciences ◽

10.3390/brainsci10080536 ◽

2020 ◽

Vol 10 (8) ◽

pp. 536

Author(s):

Gaurav Patil ◽

Patrick Nalepka ◽

Rachel W. Kallen ◽

Michael J. Richardson

Keyword(s):

Fixed Point ◽

Dynamical Systems ◽

Limit Cycle ◽

Nonlinear Dynamical Systems ◽

Rhythmic Movements ◽

Human Actions ◽

Nonlinear Dynamical ◽

Behavioral Transitions ◽

Small Set ◽

Low Dimensional

Most human actions are composed of two fundamental movement types, discrete and rhythmic movements. These movement types, or primitives, are analogous to the two elemental behaviors of nonlinear dynamical systems, namely, fixed-point and limit cycle behavior, respectively. Furthermore, there is now a growing body of research demonstrating how various human actions and behaviors can be effectively modeled and understood using a small set of low-dimensional, fixed-point and limit cycle dynamical systems (differential equations). Here, we provide an overview of these dynamical motorprimitives and detail recent research demonstrating how these dynamical primitives can be used to model the task dynamics of complex multiagent behavior. More specifically, we review how a task-dynamic model of multiagent shepherding behavior, composed of rudimentary fixed-point and limit cycle dynamical primitives, can not only effectively model the behavior of cooperating human co-actors, but also reveals how the discovery and intentional use of optimal behavioral coordination during task learning is marked by a spontaneous, self-organized transition between fixed-point and limit cycle dynamics (i.e., via a Hopf bifurcation).

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