International Journal of Bifurcation and Chaos

This paper presents a rigorous and comprehensive nonlinear circuit-theoretic foundation for the memristive Hodgkin–Huxley Axon Circuit model. We show that the Hodgkin–Huxley Axon comprises a potassium ion-channel memristor and a sodium ion-channel memristor, along with some mundane circuit elements. From this new perspective, many hitherto unresolved anomalous phenomena and paradoxes reported in the literature are explained and clarified. The yet unknown nonlinear dynamical mechanisms which give birth to the action potentials remain hidden within the memristors, and the race is on for uncovering the ultimate truth.

On April 20, 2010, an oil well cap explosion below the Deepwater Horizon, an off-shore oil rig in the Gulf of Mexico, started the worst human-caused submarine oil spill ever. Though an historic tragedy for the marine ecosystem, the unprecedented monitoring of the spill in real time by satellites and increased modeling of the natural oceanic flows has provided a wealth of data, allowing analysis of the flow dynamics governing the spread of the oil. In this work, we present the results of two computational analyses describing the mixing, mass transport, and flow dynamics of the oil dispersion in the Gulf of Mexico over the first 100 days of the spill. Transfer operator methods are used to determine the spatial partitioning of regions of homogeneous dynamics into almost-invariant sets, and Finite Time Lyapunov Exponents are used to compute pseudo-barriers to the mixing of the oil between these regions. The two methods give complementary results, generating a comprehensive description of the oil flow dynamics over time.

A neuro-fuzzy based model is proposed in this paper for estimating the Lyapunov exponents of an unknown dynamical system according solely to a set of observations. Several approaches have been presented in recent years; most of them using the approximation of both the function of the trajectory of observations and the partial derivatives, to yield the Jacobian matrix of the function. The Jacobian matrix is then employed in the Jacobian-based methods that extract the Lyapunov exponents by QR-decomposition. While the accurate estimation of Lyapunov exponents has been sought, most of the related papers mainly focus on the accuracy of the trajectory approximation. In this paper, an Adaptive Neuro-Fuzzy Inference System is presented and stated to be an efficient tool for such a purpose. Structural parameters of the proposed model as the embedding dimension and the delay time are calculated by the Takens theorem and autocorrelation function, respectively. Model validation is performed by cross approximate entropy. Then, the promising performance of the proposed model as an accurate estimation of the Lyapunov exponents and its robustness to the measurement noise are finally evaluated.

Three methods of nonlinear time series analysis, Lempel–Ziv complexity, prediction error and covariance complexity were employed to distinguish between the electroencephalograms (EEGs) of normal children, children with mild autism, and children with severe autism. Five EEG tracings per cluster of children aged three to seven medically diagnosed with mild, severe and no autism were used in the analysis. A general trend seen was that the EEGs of children with mild autism were significantly different from those with severe or no autism. No significant difference was observed between normal children and children with severe autism. Among the three methods used, the method that was best able to distinguish between EEG tracings of children with mild and severe autism was found to be the prediction error, with a t-Test confidence level of above 98%.

This paper aims to analyze and understand the irregularity and complexity of earthquake ground motions from the perspective of nonlinear dynamics. Chaotic dynamics theory and chaotic time series analysis are suggested to examine the nonlinear dynamical characteristic of strong earthquake ground motions. Based on the power spectral analysis, principal component analysis and modified false nearest neighbors method, it is illustrated qualitatively that the acceleration time series of earthquake ground motions exhibit chaotic property. Next, the chaotic time series analysis is proposed to calculate quantitatively the nonlinear characteristic parameters of acceleration time histories of near-fault ground motions. Numerical results show that the correlation dimension of these ground motions is fractal dimension. Their Kolmogorov entropy is a limited positive value, and their maximal Lyapunov exponent is larger than 0. It is demonstrated that the strong earthquake ground motions present the chaotic property rather than the pure random signals, and the severe irregularity and complexity of ground motions are the reflection of high nonlinearity of earthquake physical process.

In this paper, the authors consider the two-dimensional Navier–Stokes equations defined on Ω = ℝ × (-L, L) and , where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ω_m → Ω as m → ∞. Let and be the global attractors of the equations corresponding to Ω and Ω_m, respectively, we establish that for any neighborhood of , the global attractor enters if m is large enough.

The Lyapunov exponent gives a measure of the mean decay/growth rates of the flows of nonlinear systems. However, the Lyapunov exponent needs an infinite time interval of flows and the Jacobian matrix of system dynamics. In this paper, we propose an instantaneous decay/growth rate that is a kind of generalized Lyapunov exponent and call the instantaneous Lyapunov exponent (ILE) with respect to a decay function. The instantaneous Lyapunov exponent is one of the measures that estimate the decay and growth rates of flows of nonlinear systems by assigning a comparison function and can apply a stable system whose decay rate is slower than an exponential function. Moreover, we propose a synchronization measure of two signals using the ILE.

In this paper, a very simple criterion has been established for checking the crossing direction of a characteristic root crossing the imaginary axis for a class of time-delay systems with delay-dependent parameters, an important topic both in stability analysis and Hopf bifurcation. The criterion uses two easy-to-obtain functions with clear physical meaning from the critical stability condition and it generalizes a popular result in the literature.

A pulse skipping modulation (PSM) technique is useful for improving the efficiency of a dc–dc converter under light load conditions. Under such condition, the discrete-time model of the converter could be modeled as a discontinuous map, comprising either 1D functions only or a combination of 1D and 2D functions. Using the PSM mechanism, efficiency and spectral composition could be further improved by incorporating a voltage feedback loop. In this paper we show that, in a converter under voltage controlled PSM, the entire map can be restricted to one dimension for a wide range of parameters, by properly designing the controller. As a parameter is varied, the periodicity of the output waveform may change either monotonically or non-monotonically depending on the character of the voltage mode control loop. Technically it is desirable to operate the converter in such a way that a variation of a parameter can cause only incremental change in the periodicity and hence in the spectral composition. In the nonlinear dynamics literature, such a bifurcation scenario is known as pure period incrementing cascade. Using the theory for 1D discontinuous maps, we obtain the conditions on the converter's parameters such that such a cascade is obtained, and the variation of a parameter does not cause large variation of the periodicity. The results have been experimentally validated.

The anti-jamming scheme is an important issue in the modern communication system, where the spread-spectrum signals are usually adopted. In a complex electromagnetic environment where the jamming sources may be close to the communication devices, a large jamming-to-signal ratio is possible. How to achieve the desired accuracy for the communication system in the presence of strong jamming is a crucial and outstanding problem. To solve this problem, we propose a method to identify the desired signal and the nonstationary jamming signal via Lagrangian Coherent Structures (LCS), and a new algorithm for calculating the maximum Finite-Time Lyapunov Exponent (FTLE) is presented. As opposed to the current approach which requires a vector field in the state space at each instant time, the proposed algorithm is performed on the vector fields which are respectively built on the desired signal and the nonstationary jamming signal via Hilbert Transform. A simulation test is conducted on a binary offset carrier (BOC) modulated signal and a chirp signal. The results indicate that the proposed method can provide new features to efficiently identify the nonstationary jamming signal.

For the (2 + 1)-dimensional mKP equation, what is the dynamical behavior of its traveling wave solutions and how does it depend on the parameters of the systems? This paper will answer these questions by using the methods of dynamical systems. Thirteen exact explicit parametric representations of the traveling wave solutions are given. Bifurcations of phase portraits of the corresponding singular traveling wave system are shown.

In this study, the correlation sum and the correlation integral for chaotic time series using the Supremum norm and the Euclidean norm are discussed. The correlation integrals are then used to develop governing equations for the correlation sum, noise level and correlation dimension in which the correlation dimension and the noise level are linearly dependent on each other. Some linear estimation methods for the noise level are then introduced by using these equations. The estimation methods are applied to four chaotic time series (two artificial and two real-world). By comparing the performances of the estimations of the noise level, the best estimating method is then suggested.

This paper introduces a combinational feature extraction approach to improve speech recognition systems. The main idea is to simultaneously benefit from some features obtained from nonlinear modeling applied to speech reconstructed phase space (RPS) and typical Mel frequency Cepstral coefficients (MFCCs) which have a proved role in speech recognition field. With an appropriate dimension, the reconstructed phase space of speech signal is assured to be topologically equivalent to the dynamics of the speech production system, and could therefore include information that may be absent in linear analysis approaches. In the first part of this paper the application of Lyapunov Exponents (LE) and Fractal Dimension as two usual chaotic features in speech recognition are tested and then a short discussion is made on the weakness of these features in speech recognition. In the following, a statistical modeling approach based on Gaussian mixture models (GMMs) is applied to speech RPS. A final pruned feature set is obtained by applying an efficient feature selection approach to the combination of the parameters of the GMM model and MFCC-based features. A hidden Markov model-based (HMM) speech recognition system and TIMIT speech database are used to evaluate the performance of the proposed feature set by conducting isolated and continuous speech recognition experiments. In final Continuous Speech Recognition (CSR) Experiments, using tri-phone models, 3.7% absolute phoneme recognition accuracy improvement against using MFCC features alone were obtained.

This paper presents a detailed analysis on the dynamics of a ring network with short-cut. We first investigate the absolute synchronization on the basis of Lyapunov stability approach, and then discuss the linear stability of the trivial solution by analyzing the distribution of zeros of the characteristic equation. Based on the equivariant branching lemma, we not only obtain the existence of primary steady state bifurcation but also analyze the patterns and stability of the bifurcated nontrivial equilibria. Moreover, by means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of connection strength on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. In particular, we further consider the secondary bifurcation of the nontrivial equilibria. These studies show that short-cut may be used as a simple but efficient switch to control the dynamics of a system.

A nonlinear recurrence involving a piecewise constant McCulloch–Pitts function and two 2-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0⁺ to +∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. We show that there are four steady state solutions and that all solutions will tend to one of them. We hope that our results will be useful in further investigating neural networks involving the McCulloch–Pitts function with threshold and more general periodic coefficients.

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position.

In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.

In this paper, we mainly discuss Hopf bifurcation for planar nonsmooth general systems and Liénard systems with foci of parabolic–parabolic (PP) or focus–parabolic (FP) type. For the bifurcation near a focus, when the focus is kept fixed under perturbations we prove that there are at most k limit cycles which can be produced from an elementary weak focus of order 2k + 2 (resp.k + 1)(k ≥ 1) if the focus is of PP (resp. FP) type, and we present the conditions to ensure these upper bounds are achievable. For the bifurcation near a center, the Hopf cyclicicy is studied for these systems. Some interesting applications are presented.

This paper deals with the H_∞ synchronization problem of switched chaotic systems accompanied by a time-driven switching law and its application to secure communications. Based on the Lyapunov stability theory and linear matrix inequality (LMI) and linear matrix equality (LME) optimization techniques, an output feedback controller that guarantees the synchronization of switched master-slave chaotic systems is designed. A chaotic encryption technique that uses synchronization is proposed for securely transmitting a message over public channels. Numerical simulations of both analog and digital security communication systems are conducted to demonstrate the effectiveness of the proposed methods.

Coupled positive and negative feedback loops form an essential building block of cellular signaling pathways, but the dynamics of such a system remain to be fully explored. Here, we systematically analyze a two-component circuit with interlinked positive and negative feedback loops, focusing on feedback-induced dynamics and their mechanisms. We show that feedbacks can induce monostability, oscillation, and excitability as well as the coexistence of two attractors (including that of two different stable steady states (called Type-I bistability) and that of both a stable steady state and a stable limit cycle (called Type-II bistability)). In particular for Type-II bistability, we find that feedback-controlled molecular noise can induce stochastic switching between two different attractors, and that the first passage time between them exhibits a multi-peak distribution. These investigations provide insights for understanding the biological functions of coupled positive and negative feedback circuits from the viewpoint of dynamics.

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

This paper is a continuation of our previous work on an HIV-1 therapy model of fighting a virus with another virus [Jiang et al., 2009]. The work in [Jiang et al., 2009] investigated cascading bifurcations between equilibrium solutions, as well as Hopf bifurcation from a double-infected equilibrium solution. In this paper, we propose a modification of the model in [Revilla & Garcia-Ramos, 2003; Jiang et al., 2009] by adding a constant η to the recombinant virus equation, which accounts for the treatment of constant injection of recombinants. We study the dynamics of the new model and find that η plays an important role in the therapy. Unlike the previous model without injection of recombinant, which has three equilibrium solutions, this new model can only allow two biologically meaningful equilibrium solutions.

It is shown that there is depending on η, such that the HIV free equilibrium solution is globally asymptotically stable when the basic reproduction ratio, ; becomes unstable when . In the latter case, there occurs the double-infection equilibrium solution, , which is stable when for some larger than , and loses its stability when passes the critical value and bifurcates into a family of limit cycles through Hopf bifurcation. Our results show that appropriate injection rate can help eliminate the HIV virus in the sense that the HIV free equilibrium can be made globally asymptotically stable by choosing η > 0 sufficiently large. This is in contrast to the conclusion for the case with η = 0 in which, the recombinants do not help eliminate the HIV virus but only help reduce the HIV load in the long term sense.

Applying the averaging theory of first, second and third order to one class generalized polynomial Liénard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit.

We study curious features appearing in three systems of two coupled rings of cells with Z₃ × Z₅ exact symmetry, and Z₃ and Z₅ interior symmetries. This study was motivated by previous work by Antoneli et al. [2008, 2010], on two rings of cells coupled through a "buffer" cell, with Z₃ × Z₅ and D₃ × D₅ exact and interior symmetry groups. There, quasi-periodic behavior was found through a sequence of Hopf bifurcations, and relaxation oscillation phenomena appearing further away from the third Hopf point seemed to be necessary for a curious dynamic feature to occur. In this paper, we analyze the dynamical behavior of three distinct networks of two coupled rings of cells, with no "buffer" cell, and find an analogous bifurcation scenario to explain the appearance of quasi-periodic motion in these systems and the curious exotic feature. We compute the relevant states numerically.

In this paper, Chicone's method has been extended for perturbation systems with three parameters. By applying Diliberto's theorem and variational lemma, some conditions for continuation of periodic solutions of a three-parametric autonomous perturbation system has been introduced. Also, the continuation region in the parameter space has been found. As an application of this extension, we applied it to the Lienard system.

This paper is concerned with time-varying discrete dynamical systems in Banach spaces. A criterion of chaos induced by coupled-expansion for time-varying systems is first established and then the persistence of coupled-expansion is considered for time-varying systems under small time-varying perturbations, under which the perturbed system is shown chaotic in the strong sense of Li–Yorke. By applying this result, a map with a regular and nondegenerate snap-back repeller is shown to be still chaotic in the strong sense of Li–Yorke under small time-varying perturbations.

In this paper, the dynamical behavior of a Hamiltonian system under impulsive control is discussed by both theoretical and numerical analyses. The existence and stability of its period-one and period-three solutions are obtained in virtue of a discrete map. The conditions of existence for flip bifurcation and Hopf bifurcation are further derived by using center manifold theorem and bifurcation theory. Three attracting invariant closed curves, surrounding three fixed points respectively, are investigated. Moreover, chaos in the sense of Marotto is rigorously proven. Finally, some detailed numerical results including periodic solutions, bifurcation diagrams, and chaotic attractors, are illustrated by examples, which are in good agreement with the theoretical analysis.

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

In this study, a new autonomous chaos generator is presented via State Controlled-Cellular Neural Networks (SC-CNNs). From the numerical and circuit simulations, it has been shown that the proposed SC-CNN consisting of three cells can be easily used in chaotic applications as a chaos generator. After presenting the double scroll generation, the presented SC-CNN system is used either in multi-scroll chaotic attractor generation by adding a trigonometric function generator and or to construct a SC-CNN based chaotic masking system with feedback algorithm.

Memristors are gaining increasing interest in the scientific community for their possible applications, e.g. high-speed low-power processors or new biological models for associative memories. Due to the intrinsic nonlinear characteristic of memristive devices, it is possible to use them in the design of new dynamical circuits that are able to show complex behavior, like chaos. In this paper, two new memristive chaotic circuits are presented discussing, in particular, an approach based on Cellular Nonlinear Networks for the implementation of the memristive device. The approach investigated in this paper allows to obtain memristors with common off-the-shelf components and to observe the onset of new chaotic attractors in nonlinear circuits with memristors. Furthermore, the circuits presented in this paper, being the first examples of memristive chaotic circuits based on CNNs, can be considered as the link between the three inventions by Leon O. Chua, i.e. the memristor, the first chaotic electronic circuit and Cellular Nonlinear Networks.