A driven anharmonic oscillator is described which exhibits period doubling and chaotic behavior. The measured behavior of the oscillator under successive period doublings is in quantitative agreement with a recent theory which describes the behavior of nonlinear systems. Both the scalirg and the convergence rate predicted by the theory are verified by the experiment. The oscillator also exhibits period tripling and quintupling.

No abstract

Ordinary differential equations are often used to model the dynamics and interactions in genetic networks. In one particularly simple class of models, the model genes control the production rates of products of other genes by a logical function, resulting in piecewise linear differential equations. In this article, we construct and analyze an electronic circuit that models this class of piecewise linear equations. This circuit combines CMOS logic and RC circuits to model the logical control of the increase and decay of protein concentrations in genetic networks. We use these electronic networks to study the evolution of limit cycle dynamics. By mutating the truth tables giving the logical functions for these networks, we evolve the networks to obtain limit cycle oscillations of desired period. We also investigate the fitness landscapes of our networks to determine the optimal mutation rate for evolution.

No abstract

We present measurements of the phase diagram for a nonlinear electronic oscillator circuit which has three intrinsic frequencies. The persistence of chaos to almost zero intermode coupling and the measured structure of the chaotic zones in parameter space provide unprecedented detail in the interpretation of the Ruelle-Takens scenario for the onset of chaos.There has been a great deal of interest in the past decade in the transition to chaos in systems with a small number of intrinsic frequencies. To set the stage, we recall that the Landau ^ picture of the onset of turbulence in fluid systems involved a succession of an infinite number of Hopf bifurcations, each of which introduced a new, generally incommensurate frequency to the problem. This view was supplanted by the work of Ruelle and Takens^ (RT) and later that of Newhouse, Ruelle, and Takens,^ who proved that with as few as three independent frequencies in a dynamical system, threefrequency quasiperiodic flows are structurally unstable to the development of a strange attractor. This means that an arbitrarily small perturbation in parameter value causes a system undergoing three-frequency quasiperiodic flow to become chaotic. RT went on to speculate on this being the mechanism for the onset of turbulence in systems with many degrees of freedom.Experimental work in the ensuing years served neither to confirm the theory nor refute it because some experiments demonstrated the onset of chaos after the introduction of the third frequency, "* while others allowed the nonchaotic inclusion of three or more incommensurate frequencies.^ Grebogi, Ott, and Yorke^ showed in numerical experiments on a simple torus map system that a reasonable interpretation of these seemingly conflicting results is that though three-frequency quasiperiodicity may be structurally unstable, chaos is unlikely in a measure-theoretic sense, especially for small values of the nonlinear coupling parameter. They published some low-resolution parameter-space phase diagrams which confirmed their hypothesis, but left some question as to the exact structure of the chaotic region in the parameter space.Attracting flows on the three-torus are best characterized by the definition of two winding numbers from the system's frequencies, Q)O, (0\, and 0)2'. pi^coi/coo, pi^coilm-Clearly if neither p\ nor pi can be written as a rational number p/q with p and q integers, and pjpi is not rational, then all three frequencies are incommensurate and the flow is said to be three-frequency quasiperiodic. IfPi ^plq -^Pirh, [p.qj.s] G Z, then two of the modes are locked together, and there is two-frequency quasiperodicity. ^ If all frequencies are locked into rational ratios, then the motion is periodic. The fourth possibility is that of a strange attractor,^ which requires at least a three-dimensional phase space given the constraints that nearby trajectories diverge exponentially yet never cross, all the while the motion remaining bounded. The method of measuring a phase diagram consists of varying the parameters of...

Abstract-Relaxation oscillations exhibiting more than one time scale arise naturally from many physical systems. When relaxation oscillators are coupled in a way that resembles chemical synapses, we propose a fast method to numerically integrate such networks. The numerical technique, called the singular limit method, is derived from analysis of relaxation oscillations in the singular limit. In such limit, system evolution gives rise to time instants at which fast dynamics takes place and intervals between them during which slow dynamics takes place. A full description of the method is given for a locally excitatory globally inhibitory oscillator network (LEGION), where fast dynamics, characterized by jumping which leads to dramatic phase shifts, is captured in this method by iterative operation and slow dynamics is entirely solved. The singular limit method is evaluated by computer experiments, and it produces remarkable speedup compared to other methods of integrating these systems. The speedup makes it possible to simulate large-scale oscillator networks.

The chaotic dynamics from a nonlinear electronic circuit is shown to exhibit the universal topological structure of maps on an annulus. This suggests that the corresponding universality class is large enough to include physical systems. We suggest that low-dimensional strange attractors fall into a few classes, each characterized by distinct universal topological features.PACS numbers: 05.45.+b Several routes to chaos in dynamical systems have been proposed and analyzed in detail, the best known being the period doubling 1 and the quasiperiodic routes. 2 At the onset of chaos, such dynamical systems exhibit scaling, and certain qualitative and quantitative features of the transition are universal. 3 " 5 Many experimental systems are believed to make similar transitions to chaos, with the path determined by evaluating universal features of the transition that are unique to that path. 6,7 Some examples of universal quantities that can be used for this purpose are the generalized dimensions 4 or singularity spectra 5 of the attracting set and the trajectory scaling function 3 of the orbit at the onset.Much less is known about dynamical systems beyond the onset of chaos. Chaotic experimental systems relax to regions of phase space with very complex structure and zero measure. Generally such objects are fractal, and have structure on ever smaller length scales. They are called strange attractors, and are characterized by metric invariants (which do not change under smooth transformations) such as fractal dimension and Lyapunov exponent. However, they are not universal, and hence do not yield much detailed information about the experimental system: e.g., how to model the dynamics.An alternative class of invariants that can be used to characterize a strange attractor is the set of all periodic orbits. 8 " 10 Periodic orbits are topological invariants; i.e., any change of coordinates will not change the periodicity of an orbit. Thus changing the point of observation or the variable that is being observed in an experiment will not change the cycle structure. This is important since the characterization of the attractor should be robust.The significance of the cycles is seen from the following observations. First, since the strange attractor is the set of points of the phase space visited by the orbit after the transients have settled down, motion on it is ergodic. Thus the orbit of any point P on the strange attractor will make arbitrarily close returns to P. Because of the smoothness and nonlinearity of the dynamics, one should in general be able to move P by a small amount so that the close return becomes exact: i.e., there is a periodic point arbitrarily close to P. This means that the periodic points are dense on the strange attractor. Since the motion on the attractor is chaotic, these cycles have to be unstable.The second important observation is that the structure of the strange attractor in the neighborhood of a periodic point and the motion of points in this neighborhood are determined by the tangent space of the...

Chaos control techniques exploit the sensitivity of chaos to initial conditions by applying feedback perturbations to an accessible system parameter. Most methods apply only one perturbation per period and are thus susceptible to control failure when applied to highly unstable systems. Here we extend a recently developed model-independent, quasicontinuous chaos control technique to stabilize a high-dimensional chaotic system: the driven double pendulum.