The transition to chaos is a fundamental and widely studied problem in deterministic nonlinear dynamics. Well known routes to chaos, which include the period-doubling bifurcation route, the intermittency route, the quasiperiodic route, and the crisis route, describe transitions to low-dimensional chaotic attractors with one positive Lyapunov exponent. Transitions to high-dimensional chaotic attractors with multiple positive Lyapunov exponents have just started being addressed. In stochastic systems, transitions to chaotic-like behavior are less well characterized. Global analysis coupled with stochastic transport probability can explain emergent behavior in which stable and unstable manifolds may interact with noise to cause "stochastic chaos". Another stochastic route may induce chaotic signatures through a dimension changing bifurcation, whereby the topological dimension changes when the amplitude of the noise goes beyond a critical parameter.
In this paper we present a theory of how the Lyapunov exponents may scale with the noise amplitude in general systems. A physical class of multiscale dynamical systems will be presented to show that noise may induce low dimensional chaos, or for other parameters, may induce chaos that bifurcates to an attractor contained in a high topological dimension. We present a numerical bifurcation analysis of the resulting system, illustrating the mechanism for the onset of high dimensional chaos. By computing the constrained invariant sets, we reveal the transition from low dimensional to high dimensional chaos. Applications include both deterministic and stochastic bifurcations.
KEYWORDS: Stochastic processes, Data modeling, Gas lasers, Biology, Systems modeling, Physics, Carbon monoxide, Analytical research, Laser optics, Laser development
Many diseases, such as childhood diseases, dengue fever, and West Nile virus, appear to oscillate randomly as a function of seasonal environmental or social changes. Such oscillations appear to have a chaotic bursting character, although it is still uncertain how much is due to random fluctuations. Such bursting in the presence of noise is also observed in driven lasers. In this talk, I will show how noise can excite random outbreaks in simple models of seasonally driven outbreaks, as well as lasers. The models for both population dynamics will be shown to share the same class of underlying topology, which plays a major role in the cause of observed stochastic bursting.
In a previous paper we have introduced a new continuation method which does not require an analytical model, but only an experimental time series. Using a predictor-corrector technique the method tracks an unstable orbit of a map, through different bifurcation regimes by varying an accessible system parameter. In this method the continuation parameter was varied deterministically. That is, the location of the parameter at each continuation step is chosen by the experimenter. In this paper we introduce a similar algorithm, but now the parameter is varied randomly. We will refer to this algorithm as random-walk control or stochastic tracking. This algorithm is useful to experimentalists for canceling the effect of drift in experiments, which is always inevitable at some level.
This paper highlights some of the new contributions nonlinear dynamics has made in the areas of control and tracking. In particular, emphasis is placed on the model independent approach to control and tracking: The connections between the classical control and the control based on time series embedding methods are made. In experiments of control, our approach does not necessarily imply new equipment is needed in the loop. Rather, it is the control settings which are constructed off-line so that location of the control point and gain are determined without trial and error. Using the model independent approach also allows one to locate and control many other accessible unstable phenomena without having to construct a global nonlinear model. Tracking gives a constructive approach to control inaccessible states, as well as maps out the global regions of phase space.
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