Showing papers in "International Journal of Bifurcation and Chaos in 2009"
TL;DR: By modifying the characteristics of nonlinear memristors, the memristor DTCNN can perform almost all functions of Memristor cellular automaton and can perform more than one function at the same time, that is, it allows multitasking.
Abstract: In this paper, we design a cellular automaton and a discrete-time cellular neural network (DTCNN) using nonlinear passive memristors. They can perform a number of applications, such as logical operations, image processing operations, complex behaviors, higher brain functions, etc. By modifying the characteristics of nonlinear memristors, the memristor DTCNN can perform almost all functions of memristor cellular automaton. Furthermore, it can perform more than one function at the same time, that is, it allows multitasking.
249 citations
TL;DR: In this article, it was shown that both Voronoi diagram and its dual graph Delaunay triangulation are simultaneously constructed in cultures of plasmodium, a vegetative state of Physarum polycephalum.
Abstract: We experimentally demonstrate that both Voronoi diagram and its dual graph Delaunay triangulation are simultaneously constructed — for specific conditions — in cultures of plasmodium, a vegetative state of Physarum polycephalum. Every point of a given planar data set is represented by a tiny mass of plasmodium. The plasmodia spread from their initial locations but, in certain conditions, stop spreading when they encounter plasmodia originated from different locations. Thus space loci not occupied by the plasmodia represent edges of Voronoi diagram of the given planar set. At the same time, the plasmodia originating at neighboring locations form merging protoplasmic tubes, where the strongest tubes approximate Delaunay triangulation of the given planar set. The problems are solved by plasmodium only for limited data sets, however the results presented lay a sound ground for further investigations.
141 citations
TL;DR: This work presents a novel computational framework based on polychronous wavefront dynamics based on temporal and spatial patterns of activity in pulse-propagating media and their interaction with transponders, which create pulses in response to receiving appropriate inputs.
Abstract: There is great interest in methods for computing that do not involve digital machines. Many computational paradigms were inspired by brain research, such as Boolean neuronal logic [McCulloch & Pitts, 1943], the perceptron [Rosenblatt, 1958], attractor neural networks [Hopfield, 1982] and cellular neural nets [Chua & Yang, 1988]. All these paradigms abstract biological circuits to artificial neural networks, i.e. interconnected units (neurons) that perform computations based on the connections between the units (synapses). Here we present a novel computational framework based on polychronous wavefront dynamics. It is entirely different from an artificial neural network paradigm, rather it is based on temporal and spatial patterns of activity in pulse-propagating media and their interaction with transponders, which create pulses in response to receiving appropriate inputs, e.g. two coincident input pulses. A pulse propagates as a circular wave from its source to other transponders. Computations result from interactions between transponders, and they are encoded by the exact physical locations of transponders and by precise timings of pulses. We illustrate temporal pattern recognition, reverberating memory, temporal signal analysis and basic logical operations using polychronous wavefront computations. This work reveals novel principles for designing nanoscale computational devices.
106 citations
TL;DR: It is considered that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin.
Abstract: In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. The results are applied to a quadratic system with cubic perturbations to show that the system can have five limit cycles in the vicinity of the origin.
85 citations
TL;DR: It is revealed that even for a random-like chaotic trajectory, which is extremely sensitive to initial conditions and is long-term unpredictable, the system dynamics of a nonlinear chaotic system can still be locally-accurate identified along the chaotic trajectory in a deterministic way.
Abstract: In this paper, we investigate the problem of identifying or modeling nonlinear dynamical systems undergoing periodic and period-like (recurrent) motions. For accurate identification of nonlinear dynamical systems, the persistent excitation condition is normally required to be satisfied. Firstly, by using localized radial basis function networks, a relationship between the recurrent trajectories and the persistence of excitation condition is established. Secondly, for a broad class of recurrent trajectories generated from nonlinear dynamical systems, a deterministic learning approach is presented which achieves locally-accurate identification of the underlying system dynamics in a local region along the recurrent trajectory. This study reveals that even for a random-like chaotic trajectory, which is extremely sensitive to initial conditions and is long-term unpredictable, the system dynamics of a nonlinear chaotic system can still be locally-accurate identified along the chaotic trajectory in a deterministic way. Numerical experiments on the Rossler system are included to demonstrate the effectiveness of the proposed approach.
70 citations
TL;DR: This work shows the experimental implementation of a chaotic communication system based on two Chua's oscillators which are synchronized by Hamiltonian forms and observer approach and the suitability of the CCII+ to implement chaotic communication systems.
Abstract: This work shows the experimental implementation of a chaotic communication system based on two Chua's oscillators which are synchronized by Hamiltonian forms and observer approach. The chaotic communication scheme is realized by using the commercially available positive-type second generation current conveyor (CCII+), which is included into the AD844 device. As a result, experimental measurements are provided to demonstrate the suitability of the CCII+ to implement chaotic communication systems.
70 citations
TL;DR: It is demonstrated that bistability between the stable fixed point and limit cycle oscillations occurs for sufficiently large delay times τ and coupling strength C and as the mechanism for these delay-induced oscillations, a saddle-node bifurcation of limit cycles is identified.
Abstract: We study the nonlinear dynamics of two delay-coupled neural systems each modeled by excitable dynamics of FitzHugh–Nagumo type and demonstrate that bistability between the stable fixed point and limit cycle oscillations occurs for sufficiently large delay times τ and coupling strength C. As the mechanism for these delay-induced oscillations, we identify a saddle-node bifurcation of limit cycles.
64 citations
TL;DR: This letter presents a new hyperchaotic system, which was obtained by adding a nonlinear quadratic controller to the first equation of the three-dimensional autonomous modified Lorenz chaotic system, and which can generate very complex strange attractors with three positive Lyapunov exponents.
Abstract: This letter presents a new hyperchaotic system, which was obtained by adding a nonlinear quadratic controller to the first equation and a linear controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system. This system uses only two multipliers but can generate very complex strange attractors with three positive Lyapunov exponents. The system is not only demonstrated by numerical simulations but also implemented via an electronic circuit, showing very good agreement with the simulation results.
64 citations
TL;DR: This tutorial paper presents a history of Smale horseshoe and an overview of the progress of topological horseshOE theory, and offers a pedagogical exposition of elements of topology horseshoes theory with a lot of examples.
Abstract: In this tutorial paper, we present a history of Smale horseshoe and an overview of the progress of topological horseshoe theory. Then we offer a pedagogical exposition of elements of topological horseshoe theory with a lot of examples. Finally we demonstrate some typical applications of topological horseshoe theory to practical dynamical systems.
63 citations
TL;DR: In this paper, the most commonly used mutual information estimators, based on histograms of fixed or adaptive bin size, k-nearest neighbors and kernels, are compared and the optimization of parameters is assessed by quantifying the deviation of the estimated mutual information from its true or asymptotic value as a function of the free parameter.
Abstract: We study some of the most commonly used mutual information estimators, based on histograms of fixed or adaptive bin size, k-nearest neighbors and kernels and focus on optimal selection of their free parameters. We examine the consistency of the estimators (convergence to a stable value with the increase of time series length) and the degree of deviation among the estimators. The optimization of parameters is assessed by quantifying the deviation of the estimated mutual information from its true or asymptotic value as a function of the free parameter. Moreover, some commonly used criteria for parameter selection are evaluated for each estimator. The comparative study is based on Monte Carlo simulations on time series from several linear and nonlinear systems of different lengths and noise levels. The results show that the k-nearest neighbor is the most stable and less affected by the method-specific parameter. A data adaptive criterion for optimal binning is suggested for linear systems but it is found to be rather conservative for nonlinear systems. It turns out that the binning and kernel estimators give the least deviation in identifying the lag of the first minimum of mutual information from nonlinear systems, and are stable in the presence of noise.
58 citations
TL;DR: The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications and to economic models in which the dynamics are defined only implicitly.
Abstract: This article describes a method — called here "the method of Stretching Along the Paths" (SAP) — to prove the existence of chaotic sets in discrete-time dynamical systems. The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications. The paper provides a description of the basic mathematical ideas behind the method, as well as three applications to economic models. Incidentally, the paper also discusses some questions concerning the definition of chaos and some problems arising from economic models in which the dynamics are defined only implicitly.
TL;DR: A remarkable finding of the Letter is that hyperchaos occurs in the fractional Rossler system with order as low as 3.12, which represents the lowest order reported in literature for any hyperchaotic system studied so far.
Abstract: This Letter analyzes the hyperchaotic dynamics of the fractional-order Rossler system from a time-domain point of view. The approach exploits the Adomian decomposition method (ADM), which generates series solution of the fractional differential equations. A remarkable finding of the Letter is that hyperchaos occurs in the fractional Rossler system with order as low as 3.12. This represents the lowest order reported in literature for any hyperchaotic system studied so far.
TL;DR: A simplified Lorenz system with one bifurcation parameter is investigated by a detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bIfurcations and routes to chaos.
Abstract: A simplified Lorenz system with one bifurcation parameter is investigated by a detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations and routes to chaos. The results show that this system has complex dynamics with interesting characteristics.
TL;DR: It is shown that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractsor or two limit cycle attractors.
Abstract: Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.
TL;DR: It is derived that, if Card(End(X)) < c, then f is chaotic in the sense of Devaney if and only if f is transitive.
Abstract: Let X be a dendrite and f : X → X be a continuous map. Denote by R(f) and P(f) the sets of recurrent points and periodic points of f respectively. In this paper we show that, if the cardinal number Card(End(X)) of the set of endpoints of X is less than the cardinal number c of the continuum, then . From this we derive that, if Card(End(X)) < c, then f is chaotic in the sense of Devaney if and only if f is transitive.
TL;DR: The onset and vanishing conditions of stick motions are developed, and the condition for maintaining the stick motion is achieved as well as the corresponding physics interpretation is given for a better understanding of nonlinear behaviors of gear transmission systems.
Abstract: In this paper, an investigation on nonlinear dynamical behaviors of a transmission system with a gear pair is conducted. The transmission system is described through an impact model with a possible stick between the two gears. From the theory of discontinuous dynamical systems, the motion mechanism of impacting chatter with stick is investigated. The onset and vanishing conditions of stick motions are developed, and the condition for maintaining the stick motion is achieved as well. The corresponding physics interpretation is given for a better understanding of nonlinear behaviors of gear transmission systems. Furthermore, such an understanding may be very helpful to improve the efficiency of gear transmission systems.
TL;DR: It is proved that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and a large range of parameters for which there is a cycle is found.
Abstract: The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.
TL;DR: Existence of a quasi-minimal set is proved and an appropriate simulation of a chaotic attractor is presented for a nonautonomous differential equation with a pulse function, whose moments of discontinuity depend on the initial moment.
Abstract: We address a nonautonomous differential equation with a pulse function, whose moments of discontinuity depend on the initial moment. Existence of a quasi-minimal set is proved. An appropriate simulation of a chaotic attractor is presented.
TL;DR: Various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos are reported on, and recent results in this new field of research are reviewed.
Abstract: Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rossler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.
TL;DR: It is proved that it can be constructed successively a formal series such that the Lyapunov system is reduced a half- normal form and from the coefficients of the half-normal form, the LyAPunov constants of the origin are obtained.
Abstract: The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied.
TL;DR: The practical usefulness of nonlinear dynamical analysis for the design of a planar cable-supported beam is discussed, and the importance, for an engineering design, of a careful interpretation of: isola bifurcation, transition to chaos both by period doubling cascade and reverse boundary crisis, multistability with coexistence of chaotic and periodic attractors, fractal basins boundaries, erosion of immediate basins, interrupted sequence of period doubling bIfurcations.
Abstract: In this paper we discuss the practical usefulness of nonlinear dynamical analysis for the design of a planar cable-supported beam: we refer to a feasible case, assuming the value of the parameters corresponding to a realistic pedestrian footbridge. We consider a one degree of freedom model, obtained by the classical Galerkin reduction technique: the ensuing ordinary differential equation has both quadratic and cubic terms, due to geometric nonlinearities. Extensive numerical simulations are performed: they point out that this model, in spite of its apparent simplicity, is able to highlight the complex dynamics of the cable-supported beam, describing several common and uncommon nonlinear phenomena. Each of them is interpreted in terms of oscillations of the considered mechanical system; we explain the relevance of all the obtained results in the design of the examined structure under steady loads as wind and pedestrians, but also under transient phenomena as earthquake and gust; the ensuing issues, the most dangerous ranges and also the sensibility to perturbations are discussed in detail. In particular we deal with the importance, for an engineering design, of a careful interpretation of: isola bifurcation, transition to chaos both by period doubling cascade and reverse boundary crisis, multistability with coexistence of chaotic and periodic attractors, fractal basins boundaries, erosion of immediate basins, interrupted sequence of period doubling bifurcations. Also the effects of secondary attractors are analyzed, and it is shown that in general they cannot be neglected even if their range of existence is very small. We underline that all these investigations are performed choosing the excitation frequency far from resonances in order to alert the designer that the system dynamics may be complex independently of the activation mechanism due to resonance.
TL;DR: This paper considers both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit for a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques.
Abstract: This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the system's behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).
TL;DR: A set of ordinary differential equations is deduced which describes the nonvariational Ising–Bloch transition in unified manner and is able to explain the transition between resting and moving walls.
Abstract: Transition from motionless to moving domain walls connecting two uniform oscillatory equivalent states in both a magnetic wire forced with a transversal oscillating magnetic field and a parametrically driven damped pendula chain are studied. These domain walls are not contained in the conventional approach to these systems — parametrically driven damped nonlinear Schrodinger equation. By adding in this model higher order terms, we are able to explain these solutions and the transition between resting and moving walls. Based on amended amplitude equation, we deduced a set of ordinary differential equations which describes the nonvariational Ising–Bloch transition in unified manner.
TL;DR: This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system.
Abstract: This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in [Luo & O'Connor, 2007] are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.
TL;DR: In this article, the Bohmian quantum potential is transformed into two quantum quantum correctors, the first corrector modifies the kinetic energy term of the Hamilton-Jacobi (HJ) equation, and the second one modifies potential energy term.
Abstract: David Bohm had shown that the Schrodinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions — action and probability density. The first equation is the Hamilton–Jacobi (HJ) equation, a "visiting card" of classical mechanics, is modified by the Bohmian quantum potential. This potential is a nonlinear function of the probability density. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed into two Bohmian quantum correctors. The first corrector modifies the kinetic energy term of the HJ equation, and the second one modifies the potential energy term. The unification of the quantum HJ equation and the entropy balance equation gives a complexified HJ equation containing complex kinetic and potential terms. The imaginary parts of these terms have an order of smallness about the Planck constant. The Bohmian quantum corrector is an indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to an imaginary sector. The difference between the Bohmian and Feynman's trajectories is that the former satisfies the principle of least action and they bifurcate on interfaces. The latter covers all possible paths from a source to a detector. They can split and annihilate.
TL;DR: This work shows that the onset of synchronization takes place roughly at the same value of the order parameter as a random graph with the same size and average connectivity, and indicates that the fully synchronized state is more easily attained in random graphs.
Abstract: In this paper, we study the synchronization properties of random geometric graphs. We show that the onset of synchronization takes place roughly at the same value of the order parameter as a random graph with the same size and average connectivity. However, the dependence of the order parameter on the coupling strength indicates that the fully synchronized state is more easily attained in random graphs. We next focus on the complete synchronized state and show that this state is less stable for random geometric graphs than for other kinds of complex networks. Finally, a rewiring mechanism is proposed as a way to improve the stability of the fully synchronized state as well as to lower the value of the coupling strength at which it is achieved. Our work has important implications for the synchronization of wireless networks, and should provide valuable insights for the development and deployment of more efficient and robust distributed synchronization protocols for these systems.
TL;DR: The structure of point clouds obtained as time delay embeddings of human speech signals is studied by approximating the data sets with certain simplicial complexes and analyzing their persistent homology.
Abstract: We study the structure of point clouds obtained as time delay embeddings of human speech signals by approximating the data sets with certain simplicial complexes and analyzing their persistent homology. Results for several different sounds are presented in embedding dimensions 3 and 4. The first Betti number allows a coarse classification of sounds into three groups: vowels, nasals and noise.
TL;DR: It is shown that the Holling-type II predator–prey system with constant rate harvesting has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bIfurcation, the degenerate Bogdanov–Takens bifircation of codimension 3, the supercritical and subcritical Hopf biforcation, and the generalizedHopf b ifurcation.
Abstract: The objective of this paper is to study the dynamical properties of a Holling-type II predator–prey system with constant rate harvesting. It is shown that the model has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the degenerate Bogdanov–Takens bifurcation of codimension 3, the supercritical and subcritical Hopf bifurcation, the generalized Hopf bifurcation. These results reveal far richer dynamics than that of the model with no harvesting.
TL;DR: The alternated Julia sets are presented and it is proved analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two.
Abstract: In this work we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two. Some examples are presented.
TL;DR: The traveling wave solutions for a generalized coupled KdV equations are discussed and exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.
Abstract: By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.