Biomedical subjects

Celso Grebogi

Publications and source records attributed to Celso Grebogi.

12 recordsLinked to original sources

PubMed2003

Conditions for efficient chaos-based communication.

We find the conditions for a chaotic system to transmit a general source of information efficiently. Transmission of information with very low probability of error is possible if the topological entropy of the transmitted wave signal is greater than or equal to the Shannon entropy of the source message minus the conditional entropy coming from the limitations of the channel (such as equivocation by the noise). This condition may not be always satisfied both due to dynamical constraints and due to the nonoptimal use of the dynamical partition. In both cases, we describe strategies to overcome these limitations.

Communication↗

PubMed2002

Countable and uncountable boundaries in chaotic scattering.

We study the topological structure of basin boundaries of open chaotic Hamiltonian systems in general. We show that basin boundaries can be classified as either type I or type II, according to their topology. Let B be the intersection of the boundary with a one-dimensional curve. In type I boundaries, B is a Cantor set, whereas in type II boundaries B is a Cantor set plus a countably infinite set of isolated points. We show that the occurrence of one or the other type of boundary is determined by the topology of the accessible configuration space, and also by the chosen definition of escapes. We show that the basin boundary may undergo a transition from type I to type II, as the system's energy crosses a critical value. We illustrate our results with a two-dimensional scattering system.

Journal Article↗

PubMed2002

Chaotic bursting at the onset of unstable dimension variability.

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

Journal Article↗

PubMed2002

Rainbow transition in chaotic scattering.

We study the effects of classical chaotic scattering on the differential cross section, which is the measurable quantity in most scattering experiments. We show that the fractal set of singularities in the deflection function is not, in general, reflected on the differential cross section. We show that there are systems in which, as the energy (or some other parameter) crosses a critical value, the system's differential cross-section changes from a singular function having an infinite set of rainbow singularities with structure in all scales to a smooth function with no singularities, the scattering being chaotic on both sides of the transition. We call this metamorphosis the rainbow transition. We exemplify this transition with a physically relevant class of systems. These results have important consequences for the problem of inverse scattering in chaotic systems and for the experimental observation of chaotic scattering.

Journal Article↗

PubMed2002

Finite-size effects on active chaotic advection.

A small (but finite-size) spherical particle advected by fluid flows obeys equations of motion that are inherently dissipative, due to the Stokes drag. The dynamics of the advected particle can be chaotic even with a flow field that is simply time periodic. Similar to the case of ideal tracers, whose dynamics is Hamiltonian, chemical or biological activity involving such particles can be analyzed using the theory of chaotic dynamics. Using the example of an autocatalytic reaction, A+Bright arrow2B, we show that the balance between dissipation in the particle dynamics and production due to reaction leads to a steady state distribution of the reagent. We also show that, in the case of coalescence reaction, B+Bright arrowB, the decay of the particle density obeys a universal scaling law as approximately t(minus sign1) and that the particle distribution becomes restricted to a subset with fractal dimension D2, where D2 is the correlation dimension of the chaotic attractor in the particle dynamics.

Journal Article↗

PubMed2002

Active synchronization in nonhyperbolic hyperchaotic systems.

We propose a methodology to address the outstanding problem of synchronization in nonhyperbolic hyperchaotic physical systems. Our approach makes use of a controlling-chaos strategy that accomplishes the task by transmitting only one scalar signal even in the presence of noise.

Journal Article↗

PubMed2002

Unexpected robustness against noise of a class of nonhyperbolic chaotic attractors.

Chaotic attractors arising in physical systems are often nonhyperbolic. We compare two sources of nonhyperbolicity: (1) tangencies between stable and unstable manifolds, and (2) unstable dimension variability. We study the effects of noise on chaotic attractors with these nonhyperbolic behaviors by investigating the scaling laws for the Hausdorff distance between the noisy and the deterministic attractors. Whereas in the presence of tangencies, interactive noise yields attractor deformations, attractors with only dimension variability are robust, despite the fact that shadowing is grossly violated.

Journal Article↗

PubMed2002

Autocatalytic reactions of phase distributed active particles.

We investigate the effect of asynchronism of autocatalytic reactions taking place in open hydrodynamical flows, by assigning a phase to each particle in the system to differentiate the timing of the reaction, while the reaction rate (periodicity) is kept unchanged. The chaotic saddle in the flow dynamics acts as a catalyst and enhances the reaction in the same fashion as in the case of a synchronous reaction that was studied previously, proving that the same type of nonlinear reaction kinetics is valid in the phase-distributed situation. More importantly, we show that, in a certain range of a parameter, the phenomenon of phase selection can occur, when a group of particles with a particular phase is favored over the others, thus occupying a larger fraction of the available space, or eventually leading to the extinction of the unfavored phases. We discuss the biological relevance of this latter phenomenon. (c) 2002 American Institute of Physics.

Journal Article↗

PubMed2000

Chaotic advection, diffusion, and reactions in open flows.

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.

Journal Article↗

PubMed1997

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders.

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (c) 1997 American Institute of Physics.

Journal Article↗

PubMed1997

Multistability and the control of complexity.

We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. (c) 1997 American Institute of Physics.

Journal Article↗

PubMed1993

Conditions for the abrupt bifurcation to chaotic scattering.

One of the generic ways in which chaotic scattering can come about as a system parameter is varied is the so-called "abrupt bifurcation" in which the scattering is nonchaotic on one side of the bifurcation and is chaotic and hyperbolic on the other side. Previous work demonstrating the abrupt bifurcation [S. Bleher et al., Phys. Rev. Lett. 63, 919 (1989); Physica D 46, 87 (1990)] was primarily for the case where the scattering potential had maxima ("hilltops") which had locally circular isopotential contours. Here we extend these considerations to the more general case of locally elliptically shaped isopotential contours at the hilltops. It turns out that the conditions for the abrupt bifurcation change drastically as soon as even a small amount of noncircularity is included (i.e., the circular case is singular). The illustrative case of scattering from three isolated potential hills is dealt with in detail. One interesting result is a simple geometrical sufficient condition for an abrupt bifurcation in the case of large enough ellipticity of the hill with lowest potential at its hilltop.

Journal Article↗