Search

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincare section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.

In additive white gaussian noise (AWGN) channel, chaos has been proved to be the optimal coherent communication waveform in the sense of using very simple matched filter to maximize the signal-to-noise ratio (SNR). Recently, Lyapunov exponent spectrum of the chaotic signals after being transmitted through a wireless channel has been shown to be unaltered, paving the way for wireless communication using chaos. In wireless communication systems, inter-symbol interference (ISI) caused by multipath propagation is one of the main obstacles to achieve high bit transmission rate and low bit error rate (BER). How to resist multipath effect is a fundamental problem in a chaos-based wireless communication system (CWCS). In this paper, implementation of a CWCS is presented. It is built to transmit chaotic signals generated by a hybrid dynamical system and then to filter the received signals by using the corresponding matched filter to decrease the noise effect and to detect the binary information. We find that the multipath effect can be effectively resisted by regrouping the return map of the received signal and by setting the corresponding threshold based on the available information. We show that the optimal threshold is a function of the channel parameters and of the transmitted information symbols. Practically, the channel parameters are time-variant, and the future information symbols are unavailable. In this case, a suboptimal threshold (SOT) is proposed, and the BER using the SOT is derived analytically. Simulation results show that the CWCS achieves a remarkable competitive performance even under inaccurate channel parameters., Comment: 9 pages, 6 figures, 1 table

A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.

Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behavior, such as frequency synchronization (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, with arbitrary finite-variance distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behavior of networks.

Exploration of transient Granger causal interactions in neural sources of electrophysiological activities provides deeper insights into brain information processing mechanisms. However, the underlying neural patterns are confounded by time-dependent dynamics, non-stationarity and observational noise contamination. Here we investigate transient Granger causal interactions using source time-series of somatosensory evoked magnetoencephalographic (MEG) elicited by air puff stimulation of right index finger and recorded using 306-channel MEG from 21 healthy subjects. A new time-varying connectivity approach, combining renormalised partial directed coherence with state space modelling, is employed to estimate fast changing information flow among the sources. Source analysis confirmed that somatosensory evoked MEG was mainly generated from the contralateral primary somatosensory cortex (SI) and bilateral secondary somatosensory cortices (SII). Transient Granger causality shows a serial processing of somatosensory information, 1) from contralateral SI to contralateral SII, 2) from contralateral SI to ipsilateral SII, 3) from contralateral SII to contralateral SI and 4) from contralateral SII to ipsilateral SII. These results are consistent with established anatomical connectivity between somatosensory regions and previous source modeling results, thereby providing empirical validation of the time-varying connectivity analysis. We argue that the suggested approach provides novel information regarding transient cortical dynamic connectivity, which previous approaches could not assess.

The major fungal pathogen of humans, Candida albicans, mounts robust responses to oxidative stress that are critical for its virulence. These responses counteract the reactive oxygen species (ROS) that are generated by host immune cells in an attempt to kill the invading fungus. Knowledge of the dynamical processes that instigate C. albicans oxidative stress responses is required for a proper understanding of fungus-host interactions. Therefore, we have adopted an interdisciplinary approach to explore the dynamical responses of C. albicans to hydrogen peroxide (H2O2). Our deterministic mathematical model integrates two major oxidative stress signalling pathways (Cap1 and Hog1 pathways) with the three major antioxidant systems (catalase, glutathione and thioredoxin systems) and the pentose phosphate pathway, which provides reducing equivalents required for oxidative stress adaptation. The model encapsulates existing knowledge of these systems with new genomic, proteomic, transcriptomic, molecular and cellular datasets. Our integrative approach predicts the existence of alternative states for the key regulators Cap1 and Hog1, thereby suggesting novel regulatory behaviours during oxidative stress. The model reproduces both existing and new experimental observations under a variety of scenarios. Time- and dose-dependent predictions of the oxidative stress responses for both wild type and mutant cells have highlighted the different temporal contributions of the various antioxidant systems during oxidative stress adaptation, indicating that catalase plays a critical role immediately following stress imposition. This is the first model to encapsulate the dynamics of the transcriptional response alongside the redox kinetics of the major antioxidant systems during H2O2 stress in C. albicans.

Persistent currents (PCs), one of the most intriguing manifestations of the Aharonov-Bohm (AB) effect, are known to vanish for Schrödinger particles in the presence of random scatterings, e.g., due to classical chaos. But would this still be the case for Dirac fermions? Addressing this question is of significant value due to the tremendous recent interest in two-dimensional Dirac materials. We investigate relativistic quantum AB rings threaded by a magnetic flux and find that PCs are extremely robust. Even for highly asymmetric rings that host fully developed classical chaos, the amplitudes of PCs are of the same order of magnitude as those for integrable rings, henceforth the term superpersistent currents (SPCs). A striking finding is that the SPCs can be attributed to a robust type of relativistic quantum states, i.e., Dirac whispering gallery modes (WGMs) that carry large angular momenta and travel along the boundaries. We propose an experimental scheme using topological insulators to observe and characterize Dirac WGMs and SPCs and speculate that these features can potentially be the base for a new class of relativistic qubit systems. Our discovery of WGMs in relativistic quantum systems is remarkable because, although WGMs are common in photonic systems, they are relatively rare in electronic systems.

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.

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.

In this paper, we present a novel approach to quantify regular or chaotic dynamics of either smooth or non-smooth dynamical systems. The introduced method is applied to trace regular and chaotic stick–slip and slip–slip dynamics. Stick–slip and slip–slip periodic and chaotic trajectories are analyzed (for the investigated parameters, a stick–slip dynamics dominates). Advantages of the proposed numerical technique are given.

Chemical and biological processes often take place in fluid flows. Many of them, like environmental or microfluidical ones, generate filamentary patterns which have a fractal structure, due to the presence of chaos in the underlying advection dynamics. In such cases, hydrodynamical stirring strongly couples to the reactivity of the advected species: the outcome of the reaction is then typically different from that of the same reaction taking place in a well-mixed environment. Here we review recent progress in this field, which became possible due to the application of methods taken from dynamical system theory. We place special emphasis on the derivation of effective rate equations which contain singular terms expressing the fact that the reaction takes place on a moving fractal catalyst, on the unstable foliation of the reaction free advection dynamics.

Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high-dimensional chaos. This paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly non-hyperbolic behavior, the system undergoes on–off intermittency with respect to the synchronization manifold. This has been confirmed by numerical diagnostics of synchronization and non-hyperbolic behavior, the latter using the statistical properties of finite-time Lyapunov exponents.

We consider the dynamics of the lowest order transversal vibration mode of a suspension bridge, for which the hangers are treated as one-sided springs, according to the model of Lazer and McKeena [SIAM Review 58, 1990, 537]. We analyze in particular the multi-stability of periodic attractors and the basin of attraction structure in phase space and its dependence with the model parameters. The parameter values used in numerical simulations have been estimated from a number of bridges built in the United States and in the United Kingdom, thus taking into account realistic, yet sometimes simplified, structural, aerodynamical, and physical considerations.

Both a theoretical argument and a numerical algorithm to identify periodic and chaotic orbits are presented and discussed. Reliability of the approach is verified using the Duffing oscillator through the standard computation of Lyapunov exponents. Advantages of the proposed approach are given.

Unstable dimension variability is a mechanism whereby an invariant set of a dynamical system, like a chaotic attractor or a strange saddle, loses hyperbolicity in a severe way, with serious consequences on the shadowability properties of numerically generated trajectories. In dynamical systems possessing a variable parameter, this phenomenon can be triggered by the bifurcation of an unstable periodic orbit. This Letter aims at discussing the possible types of codimension-one bifurcations leading to unstable dimension variability in a two-dimensional map, presenting illustrative examples and displaying numerical evidences of this fact by computing finite-time Lyapunov exponents.

The time evolution of the lowest order transversal oscillation mode of a suspension bridge is studied by means of a piecewise-linear forced and damped one-dimensional oscillator, in which the loss of smoothness is due to the asymmetric response of the bridge hangers with respect to stretching and compression. If the midpoint roadbed deflection is outside a specified safe region, the bridge is supposed to collapse. We analyze the relative area of the safe basin, or the fraction of initial conditions in the phase space for which the bridge does not collapse with respect to the damping and forcing parameters. The safe basin erosion is enhanced by the appearance of incursive fingers caused by the exponential accumulation of safe basin lobes towards an invariant manifold of a periodic orbit which undergoes a homoclinic bifurcation.

Periodicity is ubiquitous in nature. In this work, we analyze the dynamical reasons for which periodic windows, that appear in parameter space diagrams, have different shapes and structures. For that, we make use of a dynamical quantity, called spine — the skeleton of the window, in order to explain a conjecture that describes the presence of periodic windows in the parameter space of high-dimensional chaotic systems.

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.

To understand, predict, and control complex networked systems, a prerequisite is to reconstruct the network structure from observable data. Despite recent progress in network reconstruction, binary-state dynamics that are ubiquitous in nature, technology, and society still present an outstanding challenge in this field. Here we offer a framework for reconstructing complex networks with binary-state dynamics by developing a universal data-based linearization approach that is applicable to systems with linear, nonlinear, discontinuous, or stochastic dynamics governed by monotonic functions. The linearization procedure enables us to convert the network reconstruction into a sparse signal reconstruction problem that can be resolved through convex optimization. We demonstrate generally high reconstruction accuracy for a number of complex networks associated with distinct binary-state dynamics from using binary data contaminated by noise and missing data. Our framework is completely data driven, efficient, and robust, and does not require any a priori knowledge about the detailed dynamical process on the network. The framework represents a general paradigm for reconstructing, understanding, and exploiting complex networked systems with binary-state dynamics.

Deprivation of essential nutrients can have stark consequences for many processes in a cell. We consider amino acid starvation, which can result in bottlenecks in mRNA translation when ribosomes stall due to lack of resources, i.e. tRNAs charged with the missing amino acid. Recent experiments also show less obvious effects such as increased charging of other (non-starved) tRNA species and selective charging of isoaccepting tRNAs. We present a mechanism which accounts for these observations, and shows that production of some proteins can actually increase under starvation. One might assume that such responses could only be a result of sophisticated control pathways, but here we show that these effects can occur naturally due to changes in the supply and demand for different resources, and that control can be accomplished through selective use of rare codons. We develop a model for translation which includes the dynamics of the charging and use of aa-tRNAs, explicitly taking into account the effect of specific codon sequences. This constitutes a new control mechanism in gene regulation which emerges at the community level, i.e., via resources used by all ribosomes., Comment: 11 pages, 5 figures

Characterizing complicated behavior from time series constitutes a fundamental problem of continuing interest and it has attracted a great deal of attention from a wide variety of fields on account of its significant importance. We in this paper propose a novel wavelet multiresolution complex network (WMCN) for analyzing multivariate nonlinear time series. In particular, we first employ wavelet multiresolution decomposition to obtain the wavelet coefficients series at different resolutions for each time series. We then infer the complex network by regarding each time series as a node and determining the connections in terms of the distance among the feature vectors extracted from wavelet coefficients series. We apply our method to analyze the multivariate nonlinear time series from our oil–water two-phase flow experiment. We construct various wavelet multiresolution complex networks and use the weighted average clustering coefficient and the weighted average shortest path length to characterize the nonlinear dynamical behavior underlying the derived networks. In addition, we calculate the permutation entropy to support the findings from our network analysis. Our results suggest that our method allows characterizing the nonlinear flow behavior underlying the transitions of oil–water flows.

Understanding the interactions among nodes in a complex network is of great importance, since they disclose how these nodes are cooperatively supporting the functioning of the network. Scientists have developed numerous methods to uncover the underlying adjacent physical connectivity based on measurements of functional quantities of the nodes states. Often, the physical connectivity, the adjacency matrix, is available. Yet, little is known about how this adjacent connectivity impacts on the "hidden" flows being exchanged between any two arbitrary nodes, after travelling longer non-adjacent paths. In this Letter, we show that hidden physical flows in conservative flow networks, a quantity that is usually inaccessible to measurements, can be determined by the interchange of physical flows between any pair of adjacent nodes. Our approach applies to steady or dynamic state of either linear or non-linear complex networks that can be modelled by conservative flow networks, such as gas supply networks, water supply networks and power grids.

An experimental secure communication method based on the Chen system with time-delay is being proposed in this paper. The Chen system with time-delay is an infinite-dimensional system having more than one positive Lyapunov exponent. The message to be transmitted is encrypted using an hyperchaotic signal generated by the Chen system with time-delay and multishift cipher function. This encryption makes difficult for an eavesdropper to reconstruct the attractor by using time-delay embedding techniques, return map reconstruction, or spectral analysis, consequently, improving the security. Simulations and experiments on TI TMS320C6713 Digital Signal Processor (DSP) show improved resilience against attack and the feasibility of the proposed scheme.

Recent years have witnessed significant interest in nanoscale physical systems, such as nanoelectromechanical and optomechanical systems, which can exhibit distinct collective dynamical behaviors, such as synchronization. As a parameter of the system changes, transition from one type of emerging collective behavior to another can occur. But what are the quantum manifestations of such a transition? We investigate a system of two optically coupled optomechanical cavities and uncover the phenomenon of transition from in-phase to antiphase synchronization. Quantum mechanically, we find that, associated with the classical transition, the entanglement measures between the various optical and mechanical degrees of freedom in the two cavities exhibit a change characteristic of second-order phase transition. These phenomena can be tested experimentally.

To search for and exploit quantum manifestations of classical nonlinear dynamics is one of the most fundamental problems in physics. Using optomechanical systems as a paradigm, we address this problem from the perspective of quantum entanglement. We uncover strong fingerprints in the quantum entanglement of two common types of classical nonlinear dynamical behaviors: periodic oscillations and quasiperiodic motion. There is a transition from the former to the latter as an experimentally adjustable parameter is changed through a critical value. Accompanying this process, except for a small region about the critical value, the degree of quantum entanglement shows a trend of continuous increase. The time evolution of the entanglement measure, e.g., logarithmic negativity, exhibits a strong dependence on the nature of classical nonlinear dynamics, constituting its signature.

In many fields of research nonlinear dynamical systems are investigated. When more than one process is measured, besides the distinct properties of the individual processes, their interactions are of interest. Often linear methods such as coherence are used for the analysis. The estimation of coherence can lead to false conclusions when applied without fulfilling several key assumptions. We introduce a data driven method to optimize the choice of the parameters for spectral estimation. Its applicability is demonstrated based on analytical calculations and exemplified in a simulation study. We complete our investigation with an application to nonlinear tremor signals in Parkinson's disease. In particular, we analyze electroencephalogram and electromyogram data.

Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable–unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

Chaotic signals can be used as carriers of information in communication systems since they own some redundancy that can be exploited to reconstruct missing or distorted parts of a waveform that has been transmitted through a communication channel. In this paper, we extend our previous results for the ideal noise free channel [Mariño et al., 1999] to a more general situation where additive white Gaussian noise corrupts the information-bearing chaotic signal.

We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the Henon map and is one that still possesses the characteristics of a Henon-type dynamics. This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the Henon map) with the map that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If e measures the degree of smoothness, we prove that, as e → 0 , the stability and the existence of the fixed points are the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.

Synchronization in chaotic systems has become an active area of research since the pioneering work of Pecora and Carroll. Most existing works, however, rely on a passive approach: A coupling between chaotic systems is necessary for their mutual synchronization. We describe here a feedback approach for synchronizing chaotic systems that is applicable in high dimensions. We show how two chaotic systems can be synchronized by applying small feedback perturbations to one of them. We detail our strategy to design the control based on the pole-placement method, and give numerical examples.

Chaotic signals can be used as carriers of information in communication systems. In this work we describe a simple encoding method that allows one to map any desired bit sequence into a chaotic waveform. The redundancy of the resulting information carrying signal enables us to devise a novel signal reconstruction technique that is able to recover relatively large parts of the chaotic signal starting from just a few samples of it. We show that this technique allows one to increase both the transmission reliability and the transmission rate of a communication system even in the presence of noise.

An aspect of the synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits unstable dimension variability. Unstable dimension variability is a cause of severe modeling difficulty for physical phenomena, since trajectories obtained from the mathematical model may not be related to trajectories of the actual system. We present and example of unstable dimension variability occurring in a system of two coupled chaotic maps, considering the dynamics in the synchronization manifold and its corresponding transversal direction, where a tongue-like structure is formed. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values., Comment: 25 pages, 8 figures, RevTex format

Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques.

We propose a communication technique that uses modeling of language in the encoding-decoding process of message transmission. A temporal partition (time-delay coarse graining of the phase space based on the symbol sequence statistics) is introduced with little if any intervention required for the targeting of the trajectory. Message transmission is performed by means of codeword, i.e., specific targeting instructions are sent to the receiver rather than the explicit message. This approach yields (i) error correction availability for transmission in the presence of noise or dropouts, (ii) transmission in a compressed format, (iii) a high level of security against undesirable detection, and (iv) language recognition.

A natural process can be deterministically modeled if solutions from its mathematical model stay close to the ones produced by nature. The mathematical model, however, is not exact due to imperfections of the natural system. We describe, in this paper, that there exists a class of models of chaotic processes, for which severe obstruction to deterministic modeling may arise. In particular, such obstruction may occur when unstable periodic orbits embedded in the chaotic invariant set have a distinct number of unstable directions, a type of nonhyperbolicity called unstable-dimension variability. We make these ideas concrete by investigating a class of deterministic models: chaotic systems with an invariant subspace such as systems of coupled chaotic oscillators. We show that unstable-dimension variability can occur in wide parameter regimes of these systems. The implications of our results to scientific modeling are discussed.

1 Telecommunication and Control Engineering Department, Polytechnic School, The University of Sao Paulo, 05508-970 Sao Paulo, Brazil 2 Laboratorio Associado de Matematica Aplicada e Computacao (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), Sao Jose dos Campos, 12227-010 Sao Paulo, Brazil 3 Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK

Systems of coupled chaotic maps and flows arise in many situations of physical and biological interest. The aim of this paper is to analyze and to present numerical evidence for a common type of nonhyperbolic behavior in these systems: unstable dimension variability. We show that unstable periodic orbits embedded in the dynamical invariant set of such a system can typically have different numbers of unstable directions. The consequence of this may be severe: the system cannot be modeled deterministically in the sense that no trajectory of the model can be realized by the natural chaotic system that the model is supposed to describe and quantify. We argue that unstable dimension variability can arise for small values of the coupling parameter. Severe modeling difficulties, nonetheless, occur only for reasonable coupling when the unstable dimension variability is appreciable. We speculate about the possible physical consequences in this case.

Previous investigations of riddling have focused on the case where the dynamical invariant set in the symmetric invariant manifold of the system is a chaotic attractor. A situation expected to arise commonly in physical systems, however, is that the dynamics in the invariant manifold is in a periodic window. We argue and demonstrate that riddling can be more pervasive in this case because it can occur regardless of whether the chaotic set in the invariant manifold is transversly stable or unstable. Scaling behavior associated with this type of riddling is analyzed and is supported by numerical experiments.

Chaotic saddles are nonattracting dynamical invariant sets that can lead to a variety of physical phenomena. We report our finding and analysis of a type of discontinuous global bifurcation (metamorphosis) of chaotic saddle that occurs in high-dimensional chaotic systems with an invariant manifold. A metamorphosis occurs when a chaotic saddle, lying in the manifold, loses stability with respect to perturbations transverse to the invariant manifold. The fractal dimension of the chaotic saddle increases abruptly through the bifurcation. We illustrate our finding by using a system of coupled maps.

This paper analyzes conditions under which dynamical systems in the plane have indecomposable continua or even infinite nested families of indecomposable continua. Our hypotheses are patterned after a numerical study of a fluid flow example, but should hold in a wide variety of physical processes. The basic fluid flow model is a differential equation in R 2 which is periodic in time, and so its solutions can be represented by a time-1 map F: R 2 → R 2 . We represent a version of this system “with noise” by considering any sequence of maps F n : R 2 → R 2 , each of which is e -close to F in the C 1 norm, so that if p is a point in the fluid flow at time n , then F n (p) is its position at time n+1 . We show that indecomposable continua still exist for small e .

Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. As a system parameter changes, chaotic saddles can evolve via an infinite number of homoclinic or heteroclinic tangencies of their stable and unstable manifolds. Based on previous numerical evidence and a rigorous analysis of a class of representative models, we show that dynamical invariants such as the topological entropy and the fractal dimension of chaotic saddles obey a universal behavior: they exhibit a devil-staircase characteristic as a function of the system parameter. {copyright} {ital 1999} {ital The American Physical Society}

A model system exhibiting a large number of attractors is investigated under the influence of noise. Several methods for discriminating two qualitatively different regions of the noise intensity are presented, and the phenomenon of noise-induced preference of attractors is reported. Finally, the relevance of our findings for detection of multiple stable states of systems occurring in nature or in the laboratory is pointed out.

A new paradigm, which combines targeting type of control problem for chaotic systems with the techniques used in system control theory, is proposed. This paradigm is used to rapidly change the evolution of a complex system among desired behaviors. We point out how this paradigm can also be applied to nonlinear systems that do not present the characteristics of a complex system.

The success of deterministic modeling of a physical system relies on whether the solution of the model would approximate the dynamics of the actual system. When the system is chaotic, situations can arise where periodic orbits embedded in the chaotic set have distinct number of unstable directions and, as a consequence, no model of the system produces reasonably long trajectories that are realized by nature. We argue and present physical examples indicating that, in such a case, though the model is deterministic and low dimensional, statistical quantities can still be reliably computed. {copyright} {ital 1999} {ital The American Physical Society}