International Journal of Bifurcation and Chaos

Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this graph as well as attractors, their basins and connecting orbits. Extensions of the framework allow, e.g. the calculation of the Morse spectrum and verification of hyperbolicity. In this work, efficient algorithms and adequate data structures will be presented for the construction of symbolic images and some basic operations on them, like the localization of the chain recurrent set and periodic orbits. The performance of these algorithms will be analyzed and we show their application in practice. The focus is not only put on several standard systems, like Lorenz and Ikeda, but also on some less well-known ones. Additionally, some tuning techniques are presented for an efficient usage of the method.

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

The unfolding of a vector field exhibiting a degenerate homoclinic orbit of inclination-flip type is studied. The linear part of the unperturbed system possesses a resonance but the coefficient of the corresponding monomial vanishes. We show that for an open set in the parameter space, the system possesses a suspended cubic Hénon-like map. As a consequence, strange attractors with entropy close to log 3 persist in a positive Lebesgue measure set.

In this paper, an oscillator with time delayed velocity feedback controls is studied in detail. Particular attention is given to internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to not only validate the analytical predictions, but also show chaotic motions for some certain parameter values.

In this paper, periodic motion in an oscillator moving on the periodically traveling belts with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions of such an oscillator are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The local stability and bifurcation for such periodic motions are obtained. The periodic motions are illustrated through the displacement, velocity and force responses in absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry-friction. The significance of this investigation lies in controlling motion of such friction-induced oscillator in industry.

We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Rényi entropy of second order K₂, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.

Recently, a new surrogate method, the Small–Shuffle (SS) surrogate method, has been proposed to investigate whether there is some kind of dynamics in irregular fluctuations, even if they are modulated by long term trends or periodicities. This situation is theoretically incompatible with the assumption underlying previously proposed surrogate methods. We apply the SS surrogate method to a variety of simulated data with known dynamics and actual time series with unknown dynamics.

The four-dimensional Hodgkin–Huxley equations describe the propagation in space and time of the action potential v(z) along a neural axon with z = x + ct and c being the pulse speed. The potential v(z), which is parameterized by the temperature, is driven by three gating functions, m(z), n(z) and h(z), each of which obeys formal first order kinetics with rate constants that are represented as nonlinear functions of the potential v. It is shown that this system can be analytically simplified (i) in the number of gating functions and (ii) in the form of associated rate functions while retaining to close approximation quantitative fidelity to computer solutions of the exact equations over the complete temperature range for which stable pulses exist. At a given temperature we record two solutions (T < T_max) corresponding to a high-speed and a low-speed branch in speed-temperature plots, c(T), or no solution (T > T_max). The pulse is considered as composed of two contiguous parts: (i) a pulse front extending from v(0) = 0 to a pulse maximum v = V_max, and (ii) a pulse back extending from V_max through a pulse minimum V_min to a final regression back to v(z → ∞) = 0. An approximate analytic solution is derived for the pulse front, which is predicted to propagate at a speed c(T) = 1203 Θ^⅜ (T°C) cm/sec, in close agreement with computer solution of the exact Hodgkin–Huxley equations for the entire pulse. These results provide the basis for a derivation of two-dimensional differential equation systems for the pulse front and pulse back, which predict the pulse maximum and minimum over the operational temperature range 0 ≤ T ≤ 25°C, in close agreement with the exact equations. Most neuron dynamics studies have been based on voltage clamp experiments featuring external current injection in place of self-generating pulse propagation. Since the behaviors of the gating functions are similar, it is suggested that the present approximations might be applicable to such situations as well as to the dynamics of myelinated fibers.

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

In this paper we study the relationships between local and global properties in networks of dynamical systems by focusing on two global properties, synchronization and peak-to-peak dynamics, and on two local properties, coherence of the components of the network and coupling strength. The analysis is restricted to networks of low-dimensional chaotic oscillators, i.e. oscillators which have peak-to-peak dynamics when they work in isolation. The results are obtained through simulation, first by considering pairs of coupled Lorenz, Rössler and Chua systems, and then by studying the behavior of spatially extended tritrophic food chains described by the Rosenzweig–MacArthur model. The conclusion is that synchronization and peak-to-peak dynamics are different aspects of the same collective behavior, which is easily obtained by enhancing local coupling and coherence. The importance of these findings is briefly discussed within the context of ecological modeling.

This paper investigates the synchronization problem for a particular class of neural networks subject to time-varying delays and input nonlinearity. Using the variable structure control technique, a memoryless decentralized control law is established which guarantees exponential synchronization even when input nonlinearity is present. The proposed controller is suitable for application in delayed cellular neural networks and Hopfield neural networks with no restriction on the derivative of the time-varying delays. A two-dimensional cellular neural network and a four-dimensional Hopfield neural network, both with time-varying delays, are presented as illustrative examples to demonstrate the effectiveness of the proposed synchronization scheme.

The following paper introduces an application of Cellular Neural Networks for the generation of predetermined stochastic textures. The key element for the task realization is an appropriate selection of template elements, which should provide a transformation of initial, random CNN state into a stable equilibrium, featuring desired perceptual properties. A template derivation procedure comprises two steps: linear CNN design, followed by a template-refinement procedure that involves nonlinear optimization. In addition, a procedure that extends CNN texture rendition capabilities into a realm of non-pure stochastic textures is proposed.

In this letter, the mosaic solutions of one-dimensional Cellular Neural Networks system (CNNs) are investigated. Three types of parameters, the synaptic weights, the input terms and the threshold are properly chosen in terms of Chua's driving-point plot. Moreover, we give sufficient conditions for the existence of the mosaic solutions.

Among reaction–diffusion systems showing Turing patterns, the diffusive Gray–Scott model [Pearson, 1993], stands out by showing self-replicating patterns (spots), which makes it the ideal simple model for developmental research. A first study of the influence of noise in the Gray–Scott model was performed by Lesmes et al. [2003] concluding that there exists an optimal noise intensity for which spot multiplication is maximal. Here we show in detail the transition from nonspotlike to spotlike pattern, with the identification of a wide range of noise intensities instead of an optimal value for which this transition occurs. Additional studies also reveal that noise produces a shift and a shrinkage of the regions of spatial patterns in the parameters space, without introducing qualitative changes to the diagram.

We study a three-state stochastic particle system on the square lattice, which extends the contact process. The phase diagram is analyzed by the mean-field approximation, the pair approximation, and numerics. The pair approximation turns out to be better than the meanfield ansatz. We also show that the Harris-FKG type correlation inequalities approximately hold in the present model.

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest nontrivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.

An approach to multivariate phase synchronization analysis in the form of a Synchronization Cluster Analysis (SCA) was introduced in [Allefeld & Kurths, 2004]. A statistical model of a synchronization cluster was described, and an abbreviated instruction on how to apply this model to empirical data was given, while an implementation of the corresponding algorithm was (and is) available from the authors. In this letter, the complete details on how the data analysis algorithm is to be derived from the model are filled in.

A system that consists of two impacting oscillators with damping has been considered in this paper. The goal of the studies was to determine the relation between the values of resonant frequencies of the oscillators and the existence of their stable periodic motion. This paper indicates various origins of the periodicity of motion and offers a some advice to the designers of systems with impacts. Especially, the results of the considerations point out some potentially dangerous consequences of the improper value of the resonant frequencies ratio.

Dynamic phenomena in a device network consisting of Dynamic Self-Electro-Optic-Effect Devices (D-SEED) are theoretically predicted and discussed via phenomenological and numerical methods, where each D-SEED element shows nonlinear oscillatory behavior. Analysis is based on a phenomenological model for photocarriers proposed in previous works by one of the authors and it treats a set of diffusively coupled nonlinear rate equations for the photocarrier density of each D-SEED.

In this paper, we report a common phenomenon observed in chaotic systems linked by time delay. Recently, the Lorenz chaotic system has been extended to the family of Lorenz systems which includes the Chen and Lü systems. These three chaotic systems, corresponding to different sets of system parameter values, are topologically different. With the aid of numerical simulations, we have surprisingly found that a simple time delay, directly applied to one or more state variables, transforms the Lorenz system to the generalized Chen system or the generalized Lü system without any parameter changes. The existence of this phenomenon has also been found in other known chaotic systems: the Rössler system, the Chua's circuit and the 4-Liu system. This finding has shown a common characteristic of chaotic systems: a new chaotic "branch" can be created from a chaotic attractor by simply adding a time delay.

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional nonautonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth one-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are well understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.

By introducing a concept of the time-angle difference for planar complex analytic systems, two recursive formulas of the function of the time-angle difference and period constants of complex isochronous centers can be given. Using these results to a class of real cubic system, the first five period constants and the isochronous center conditions of the origin are obtained. One open problem posed by [Cairo et al., 1999] is proved.

In this paper we study C¹ linearization and classification of germs of hyperbolic vector fields on Rⁿ. Two sets of algebraic conditions imposed on the eigenvalues of vector fields are given, under either of which strict similarity of the linear parts is necessary and sufficient for C¹ conjugacy of two vector fields. Furthermore, one set of the algebraic conditions together with strict similarity also imply linearization.