International Journal of Bifurcation and Chaos

Fascinating tools to understand the link between microscopic and macroscopic dynamics in many systems, lattice gases have been developed into two important directions: thermodynamic phase transitions and hydrodynamics.

Here we present how mean-field approaches allow a more thorough analysis of these systems. The kinetic equations are nonlinear and allow to consider far from equilibrium systems. Another important advantage is to gain more flexibility for practical applications.

In this paper we present the construction of stable stationary solutions in reaction–diffusion systems consisting of a 1-D array of bistable cells with a cubic nonlinearity and with a cubic-like piecewise-linear nonlinearity. Some periodic solutions, kinks, solitons are considered. While it is known that spatial chaos arises in such systems with small coupling constants, we show the existence of spatial chaos for an arbitrary value of the cell coupling constant, in the case of the piecewise-linear nonlinearity. The value of the spatial entropy is found. We also show the existence of stable spatially periodic (pattern) solutions that persist for large coupling constants.

Nonlinear two-dimensional Rayleigh–Bénard convection with periodic boundary conditions in the horizontal is studied for spatial periods near the 1:2 steady state mode interaction. The boundary conditions at the bottom are no-slip, and convection is driven by a fixed imposed temperature difference across the layer. Homotopic continuation is used to continue the boundary conditions at the top from no-slip (β=0) to stress-free (β=1). When β=0 and non-Boussinesq effects are absent the system has midplane reflection symmetry and the 1:2 resonance is weak. When β=1 this symmetry is strongly broken and the resonance is strong. The transition between these two cases is explored for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

This paper considers the behavior of a system of globally coupled identical parametric oscillators. The coupling is linear, with a phase chosen so that the system takes on steady, rather than oscillatory, states, and this work focuses on bifurcations from the zero steady state. Approximate expressions are given for the states which emerge at these bifurcations, and full expressions are given for special cases. Secondary bifurcations are also briefly discussed.

In our paper, we report the findings that the interspike intervals of the injured dorsal root ganglion neurons have multipeaked histograms and the interspike intervals are chaotic. First, the symbolic analysis method presented in the paper and the nonlinear forecasting method are used to study the dynamic differences between the interspike intervals with a multipeaked histogram generated by chaos and those induced by noise in some neuron models. Then based on the dynamic differences, the interspike intervals recorded in our experiment are analyzed by the nonlinear forecasting and the symbolic analysis method. We obtain that the dynamics of our experimental interspike intervals are greatly different from those of the interspike intervals with a multipeaked histogram induced by noise, while they are similar to the dynamics of this kind of interspike intervals generated by chaos. The results show that the experimental interspike intervals with a multipeaked histogram are chaotic.

The cubic escape equation system and the swing equation system have been widely researched because of their applicability to various physical systems and technological areas. They have confinement well structure and then have some bound solutions, which can be chaotic under periodic excitation, and a solution divergent to infinity. The inherent potential nature of these systems is described as the action variable versus the angular frequency characteristic of closed orbits of their originating Hamiltonian systems, which is important to explain the occurrence of subharmonic and harmonic resonances under the injection of external periodic driven force. This characteristic of the above systems is qualitatively same and their steady oscillational response and the basin structure, where often some attractors coexist, have some common features. The quintic potential system can also have one confinement well and the divergent solution so that it seems to exhibit similar responses under periodic excitation. However the action variable versus the angular frequency characteristic is remarkably different, which has one maximum point, while the first two systems have monotonically decreasing ones. This produces more complicated pattern of existence region of periodic points, and more various oscillational responses are observed. In this paper the author will report about the steady state oscillations and their basin structure in the quintic potential system driven by external periodic force with numerical simulations and geometrical methods of dynamical systems.

This article introduces a direct method for the numerical estimation of the spectra of Liapunov exponents based on the exterior algebra of measure elements. This approach is computationally more robust than the classical approach based on the evolution of n vectors initially possessing linear independence. Several applications of the method proposed are discussed in connection with finite-dimensional systems exhibiting hyperchaos as well as distributed parameter systems associated with reaction/diffusion kinetics.

In ecological systems, long, high-quality time series of population abundances is seldom available. However, in some cases populations are scattered over spatial domains where physical conditions appear to be homogeneous, and spatial snapshots of population abundance can be obtained. Since these spatial patterns are generated from underlying dynamical rules, an important problem is to detect the presence of determinism in these structures. In this paper, a simple model of determinism detection from spatial snapshots is presented, using a coupled map lattice formalism. It involves a simple forecasting approach where subparts of the lattice are used in order to make predictions based on the distance between neighbors. It is shown that the method is able to properly detect the presence of low-dimensional dynamics from single snapshots.

In the present work we performed an analysis of time series of instantaneous temperature and pressure produced during constant temperature molecular dynamics. Simulations were applied to a nickel oxide grain boundary for a temperature range 0.15-0.80T_m, T_m being the melting point of the system. We performed a series of analysis for these time series including test for randomness, power spectrum, Hurst exponent and structure function test. Evidence of fractional Brownian motion was found. Pressure presents 1/f over the whole range of frequencies of the system while temperature presents a two-regime behavior: white noise at low frequencies and 1/f^α at high frequencies with α increasing as a function of temperature. The origins of this behavior are discussed.

A generalized information criterion is proposed to determine an embedding dimension and a delay time for delay coordinates of the reconstructed dynamics both for linear stochastic and nonlinear deterministic processes. While the standard maximum likelihood type method requires statistical parametric models such as autoregressive models, the generalized information criterion is constructed from the quantity in accordance with the second-order Renyi entropy in terms of the correlation integral for the finite number of data which is directly obtained from a time delay vector. It is found numerically that the present method works well when applied to chaotic and stochastic systems.

Bifurcations of solitary waves and domain wall waves for a KdV-like equation with higher order nonlinearity are studied, by using bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Numbers of solitary waves and domain wall waves are given. Under some parameter conditions, a lot of explicit formulas of solitary wave solutions and domain wall solutions are obtained.

The energy increment spectrum method is developed for the numerical prediction of a specific primary resonant layer, and the width of the resonant layer can be estimated through the energy increment spectrum. This numerical approach is applied to investigate the (2M:1)-librational and (M:1)-rotational, resonant layers in a parametrically excited pendulum, and the corresponding analytical conditions for such resonant layers are developed. The numerical approach predicts the appearance and disappearance of resonant layers in nonlinear Hamiltonian systems rather than the conventional Poincaré mapping method. Illustrations of the analytical and numerical results for the appearance and disappearance of the resonant layers are given. The width of the resonant layers in the paremetric pendulum is computed. The analytical method should be further improved through renormalization.

In this note we characterize all generators of Darboux polynomials of the Rössler system by using weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations. As a corollary we prove that the Rössler system is not algebraically integrable, and that every rational first integral is a rational function in the variable x²+y²+2z. Moreover, we characterize the topological phase portrait of the Darboux integrable Rössler system.

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.

In this paper, we consider feedback control that stabilizes unstable periodic orbits (UPOs) of chaotic discrete-time systems. First, we show that there exists a strong necessary condition for stabilization of the UPOs when we use delayed feedback control (DFC) that is known as one of the useful methods for controlling chaotic systems. The condition is similar to that in the fixed point stabilization problem, in which it is impossible to stabilize the target unstable fixed point if the Jacobian matrix of the linearized system around it has an odd number of real eigenvalues greater than unity.

In order to stabilize UPOs which cannot be stabilized by the standard DFC, we adopt prediction-based control. We show a necessary and sufficient condition for the stabilization of the UPOs with arbitrary period.

A general method allowing to construct nonlinear resistors with arbitrary current–voltage (I–V) characteristics is proposed. The example of a cubic I–V characteristic is presented showing a perfect agreement between the theoretical desired resistor and its electronic realization based on analog multipliers.