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

2014 ◽  
Vol 24 (07) ◽  
pp. 1430020 ◽  
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.

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

2008 ◽  
Vol 18 (06) ◽  
pp. 1749-1758 ◽  
Author(s):  
WEN-WEN TUNG ◽  
JING HU ◽  
JIANBO GAO ◽  
VINCENT A. BILLOCK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Stable Fixed Point ◽  
Lorenz Equations ◽  
Periodic Forcing ◽  
Parameter Region ◽  
Interesting Phenomenon ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors

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.

Chaotic Oscillation via Edge of Chaos Criteria

2017 ◽  
Vol 27 (11) ◽  
pp. 1730035 ◽  
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.

Optimal tracking control of nonlinear dynamical systems

2008 ◽  
Vol 464 (2097) ◽  
pp. 2341-2363 ◽  
Author(s):  
Firdaus E Udwadia
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Weighted Norm ◽  
Analytical Dynamics ◽  
Control Cost ◽  
Nonlinear Dynamical ◽  
First Order ◽  
Time Optimal

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

Sign in / Sign up

Export Citation Format

Share Document