Biomedical subjects

Celso Grebogi

Publications and source records attributed to Celso Grebogi.

At least 19 recordsLinked to original sources

PubMed2006

Archetypal oscillator for smooth and discontinuous dynamics.

We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

Journal Article↗

PubMed2006

Dynamically multilayered visual system of the multifractal fly.

We dynamically analyze our experimental results on the motion sensitive spiking H1 neuron of the fly's visual system. We find that the fly uses an alphabet composed of a few letters to encode the information contained in the stimulus. The alphabet dynamics is multifractal both with and without stimulus, though the multifractality increases with the stimulus entropy. This is in sharp contrast to models generating independent spike intervals, whose dynamics is monofractal.

Animals↗

PubMed2006

Control of chaos and its relevancy to spacecraft steering.

In 1990, a seminal work named controlling chaos showed that not only the chaotic evolution could be controlled, but also the complexity inherent in the chaotic dynamics could be exploited to provide a unique level of flexibility and efficiency in technological uses of this phenomenon. Control of chaos is also making substantial contribution in the field of astrodynamics, especially related to the exciting issue of low-energy transfer. The purpose of this work is to bring up the main ideas regarding the control of chaos and targeting, and to show how these techniques can be extended to Hamiltonian situations. We give realistic examples related to astrodynamics problems, in which these techniques are unique in terms of efficiency related to low-energy spacecraft transfer and in-orbit stabilization.

Algorithms↗

PubMed2006

Finite-size effects on open chaotic advection.

We study the effects of finite-sizeness on small, neutrally buoyant, spherical particles advected by open chaotic flows. We show that, when observed in the configuration or physical space, the advected finite-size particles disperse about the unstable manifold of the chaotic saddle that governs the passive advection. Using a discrete-time system for the dynamics, we obtain an expression predicting the dispersion of the finite-size particles in terms of their Stokes parameter at the onset of the finite-size induced dispersion. We test our theory in a system derived from a flow and find remarkable agreement between our expression and the numerically measured dispersion.

Journal Article↗

PubMed2005

Poincaré recurrence and measure of hyperbolic and nonhyperbolic chaotic attractors.

We study Poincaré recurrence of chaotic attractors for regions of finite size. Contrary to the standard case, where the size of the recurrent regions tends to zero, the measure is no longer supported solely by unstable periodic orbits of finite length inside it, but also by other special recurrent trajectories, located outside that region. The presence of the latter leads to a deviation of the distribution of the Poincaré first return times from a Poissonian. Consequently, by taking into account the contribution of these special recurrent trajectories, a corrected estimate of the measure is obtained. This has wide experimental implications, as in the laboratory all returns can exclusively be observed for regions of finite size, and only unstable periodic orbits of finite length can be detected.

Journal Article↗

PubMed2005

Using geometric control and chaotic synchronization to estimate an unknown model parameter.

We present a new parameter estimation procedure for nonlinear systems. Such technique is based on the synchronization between the model and the system whose unknown parameter is wanted. Synchronization is accomplished by controlling the model to make it follow the system. We use geometric nonlinear control techniques to design the control system. These techniques allow us to derive sufficient conditions for synchronization and hence for proper parameter estimation. As an example, this procedure is used to estimate a parameter of an example serving as a model.

Journal Article↗

PubMed2005

Effective dynamics in Hamiltonian systems with mixed phase space.

An adequate characterization of the dynamics of Hamiltonian systems at physically relevant scales has been largely lacking. Here we investigate this fundamental problem and we show that the finite-scale Hamiltonian dynamics is governed by effective dynamical invariants, which are significantly different from the dynamical invariants that describe the asymptotic Hamiltonian dynamics. The effective invariants depend both on the scale of resolution and the region of the phase space under consideration, and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the Hamiltonian system is modeled as a chain of hyperbolic systems.

Journal Article↗

PubMed2004

Stability properties of nonhyperbolic chaotic attractors with respect to noise.

We study local and global stability of nonhyperbolic chaotic attractors contaminated by noise. The former is given by the maximum distance of a noisy trajectory from the noisefree attractor, while the latter is provided by the minimal escape energy necessary to leave the basin of attraction, calculated with the Hamiltonian theory of large fluctuations. We establish the important and counterintuitive result that both concepts may be opposed to each other. Even when one attractor is globally more stable than another one, it can be locally less stable. Our results are exemplified with the Holmes map, for two different sets of parameter, and with a juxtaposition of the Holmes and the Ikeda maps. Finally, the experimental relevance of these findings is pointed out.

Journal Article↗

PubMed2004

Reactions in flows with nonhyperbolic dynamics.

We study the reaction dynamics of active particles that are advected passively by 2D incompressible open flows, whose motion is nonhyperbolic. This nonhyperbolicity is associated with the presence of persistent vortices near the wake, wherein fluid is trapped. We show that the fractal equilibrium distribution of the reactants is described by an effective dimension d(eff) , which is a finite resolution approximation to the fractal dimension. Furthermore, d(eff) depends on the resolution epsilon and on the reaction rate 1/tau . As tau is increased, the equilibrium distribution goes through a series of transitions where the effective dimension increases abruptly. These transitions are determined by the complex structure of Cantori surrounding the Kolmogorov-Arnold-Moser (KAM) islands.

Journal Article↗

PubMed2004

Two-dimensional map for impact oscillator with drift.

An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau , relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.

Journal Article↗

PubMed2004

Chemical and biological activity in three-dimensional flows.

We study the dynamics of active particles advected by three-dimensional (3D) open incompressible flows, both analytically and numerically. We find that 3D reactive flows have fundamentally different dynamical features from those in 2D systems. In particular, we show that the reaction's productivity per reaction step can be enhanced, with respect to the 2D case, while the productivity per unit time in some 3D flows goes to zero in the limit of high mixing rates, in contrast to the 2D behavior, in which the productivity goes to a finite constant. These theoretical predictions are validated by numerical simulations on a generic map model.

Biology↗

PubMed2004

Escaping from nonhyperbolic chaotic attractors.

The noise-induced escape process from a nonhyperbolic chaotic attractor is of physical and fundamental importance. We address this problem by uncovering the general mechanism of escape in the relevant low noise limit using the Hamiltonian theory of large fluctuations and by establishing the crucial role of the primary homoclinic tangency closest to the basin boundary in the dynamical process. In order to demonstrate that, we provide an unambiguous solution of the variational equations from the Hamiltonian theory. Our results are substantiated with the help of physical and dynamical paradigms, such as the Hénon and the Ikeda maps. It is further pointed out that our findings should be valid for driven flow systems and for experimental data.

Models, Theoretical↗

PubMed2004

Reactive particles in random flows.

We study the dynamics of chemically or biologically active particles advected by open flows of chaotic time dependence, which can be modeled by a random time dependence of the parameters on a stroboscopic map. We develop a general theory for reactions in such random flows, and derive the reaction equation for this case. We show that there is a singular enhancement of the reaction in random flows, and this enhancement is increased as compared to the nonrandom case. We verify our theory in a model flow generated by four point vortices moving chaotically.

Models, Biological↗

PubMed2004

Universality in active chaos.

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treatment of the problem through partial differential equations difficult. Here we present a simple approach for the activity in inhomogeneously stirred flows. We show that the fractal patterns serving as skeletons and catalysts lead to a rate equation with a universal form that is independent of the flow, of the particle properties, and of the details of the active process. One aspect of the universality of our approach is that it also applies to reactions among particles of finite size (so-called inertial particles).

Computer Simulation↗

PubMed2003

Reactive dynamics of inertial particles in nonhyperbolic chaotic flows.

Anomalous kinetics of infective (e.g., autocatalytic) reactions in open, nonhyperbolic chaotic flows are important for many applications in biological, chemical, and environmental sciences. We present a scaling theory for the singular enhancement of the production caused by the universal, underlying fractal patterns. The key dynamical invariant quantities are the effective fractal dimension and effective escape rate, which are primarily determined by the hyperbolic components of the underlying dynamical invariant sets. The theory is general as it includes all previously studied hyperbolic reactive dynamics as a special case. We introduce a class of dissipative embedding maps for numerical verification.

Journal Article↗

PubMed2003

Why are chaotic attractors rare in multistable systems?

We show that chaotic attractors are rarely found in multistable dissipative systems close to the conservative limit. As we approach this limit, the parameter intervals for the existence of chaotic attractors as well as the volume of their basins of attraction in a bounded region of the state space shrink very rapidly. An important role in the disappearance of these attractors is played by particular points in parameter space, namely, the double crises accompanied by a basin boundary metamorphosis. Scaling relations between successive double crises are presented. Furthermore, along this path of double crises, we obtain scaling laws for the disappearance of chaotic attractors and their basins of attraction.

Models, Theoretical↗

PubMed2003

Searching in small-world networks.

We study the average time it takes to find a desired node in the Watts-Strogatz family of networks. We consider the case when the look-up time can be neglected and when it is important, where the look-up time is the time needed to choose one among all the neighboring nodes of a node at each step in the search. We show that in both cases, the search time is minimum in the small-world regime, when an appropriate distance between the nodes is defined. Through an analytical model, we show that the search time scales as N(1/D(D+1)) for small-world networks, where N is the number of nodes and D is the dimension of the underlying lattice. This model is shown to be in agreement with numerical simulations.

Journal Article↗

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↗