International Journal of Bifurcation and Chaos

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ¹. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

Visual illusion is the fallacious perception of reality or some actually existing object. In this paper, we imitate the mechanism of Ehrenstein illusion, neon color spreading illusion, watercolor illusion, Kanizsa illusion, shifted edges illusion, and hybrid image illusion using the Open Source Computer Vision Library (OpenCV). We also imitate these illusions using Cellular Neural Networks (CNNs). These imitations suggest that some illusions are processed by high-level brain functions. We next apply the morphological gradient operation to anomalous motion illusions. The processed images are classified into two kinds of images, which correspond to the central drift illusion and the peripheral drift illusion, respectively. It demonstrates that the contrast of the colors plays an important role in the anomalous motion illusion. We also imitate the anomalous motion illusions using both OpenCV and CNN. These imitations suggest that some visual illusions may be processed by the illusory movement of animations.

Particle swarm optimization (PSO) is introduced to implement a new constructive learning algorithm for training generalized cellular neural networks (GCNNs) for the identification of spatio-temporal evolutionary (STE) systems. The basic idea of the new PSO-based learning algorithm is to successively approximate the desired signal by progressively pursuing relevant orthogonal projections. This new algorithm will thus be referred to as the orthogonal projection pursuit (OPP) algorithm, which is in mechanism similar to the conventional projection pursuit approach. A novel two-stage hybrid training scheme is proposed for constructing a parsimonious GCNN model. In the first stage, the orthogonal projection pursuit algorithm is applied to adaptively and successively augment the network, where adjustable parameters of the associated units are optimized using a particle swarm optimizer. The resultant network model produced at the first stage may be redundant. In the second stage, a forward orthogonal regression (FOR) algorithm, aided by mutual information estimation, is applied to refine and improve the initially trained network. The effectiveness and performance of the proposed method is validated by applying the new modeling framework to a spatio-temporal evolutionary system identification problem.

Reversible cellular automata are discrete invertible dynamical systems determined by local interactions among their components. For the one-dimensional case, there are classical references providing a complete characterization based on combinatorial properties. Using these results and the simulation of every automaton by another with neighborhood size 2, this paper describes other types of invertible behaviors embedded in these systems different from the classical one observed in the temporal evolution. In particular, spatial reversibility and diagonal surjectivity are studied, and the generation of macrocells in the evolution space is analyzed.

A degeneration in a two-parameter setting for the focus-center-limit cycle bifurcation of symmetric 3D piecewise linear systems with three zones is studied. The analysis gives a rigorous mathematical explanation for the appearance in piecewise linear oscillators of up to two limit cycles along with a related hysteresis phenomenon.

A piecewise linear oscillator, which is a member of the generic Chua's circuit family, is shown to exhibit bistability, that is, coexistence of stable oscillations with a stable steady state. Also, the corresponding hysteresis in the appearance of periodic oscillations is explained. The analytical results included in the paper predict accurately the observed behavior.

We study the system ẋ = x(y+2z+(15/2η²)u), ẏ = y(x-2z-(7/2η²)u), ż = -z(x+y+(4/η²)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.

We present a case of a strange nonchaotic attractor (SNA) which is also nonfractal and continuous. This provides a counterexample to the widely extended assumption about the intrinsic fractal nature of any SNA. We also show that the most useful techniques to characterize the SNA fail for this case.

We studied a hybrid dynamical system composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Two typical examples of this system were investigated from the viewpoint of dynamical systems. One example is a nonfeedback system whose higher module stochastically switches inputs to the lower module. The dynamics was characterized by attractive and invariant fractal sets with hierarchical clusters addressed by input sequences. The other example is a feedback system whose higher module switches in response to the states of the lower module at regular intervals. This system converged into various switching attractors that correspond to infinite switching manifolds, which define each feedback control rule at the switching point. We showed that the switching attractors in the feedback system are subsets of the fractal sets in the nonfeedback system. The feedback system can be considered an automaton that generates various sequences from the fractal set by choosing the typical switching manifold. We can control this system by adjusting the switching interval to determine the fractal set as a constraint and by adjusting the switching manifold to select the automaton from the fractal set. This mechanism might be the key to developing information processing that is neither too soft nor too rigid.

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Z_i can be expressed as the sum of two subsystems so that Z_i = X_i + Y_i and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

In this communication the emergent behavior of clusters of robots with chaotic kinematics attractors is shown. Even if the aim of the research started from the study of the cooperative behavior of inspection robots, the reported results (including also the role of chaotic synchronization in the generation of the kinematics trajectory), show new aesthetic features of the motion in mechanical control systems.

This paper suggests a master-slave nonautonomous synchronization scheme with variable substitution control. According to the scheme, some sufficient algebraic criteria for global chaos synchronization of the master and slave nonautonomous gyrostat systems via various single-variable coupling are derived. The effectiveness of the obtained criteria is numerically verified by some examples.

This paper presents the synchronization between uncertain hyperchaotic and chaotic systems. Based on Lyapunov stability theory, an adaptive controller is derived to achieve synchronization of hyperchaotic and chaotic systems, including the case of unknown parameters in these two systems. The T.N.Č. hyperchaotic oscillator is used as the master system, and the Rössler system is used as the slave system. Numerical simulations verify these results. Additionally, the effect of noise is investigated by measuring the mean squared error (MSE) of two systems.

In this paper, we have studied the mathematical properties and their physical implications of a system of nonlinear ordinary differential equations with two variables and six parameters, which was proposed to model the martensitic transformation of shape memory alloys. While the system could not have limit cycles, bifurcation to hysteresis was found in load and/or temperature controlled processes with large enough interfacial energies. This agrees qualitatively quite well with the experimental observation and theoretical understandings of the stress-strain-temperature hysteresis of the alloys. A three-dimensional bifurcation diagram was identified. Nonhyperbolic equilibrium points were found as saddle nodes and high order node points. The local behavior was studied and the phase portrait of the system was obtained for the load and temperature parameters. Accordingly, a stable node represents the stable martensitic or austenitic phase, a saddle stands for the unstable phase mixture, and a saddle node corresponds to the beginning or the end of the transformation. Therefore, for a thermal-mechanical loading path, the martensitic transformation process accords with the qualitative changes of the equilibrium points.

In this paper, we characterize the dynamics of the Chen system ẋ = a(y - x), ẏ = (c - a)x - xz + cy, ż = xy - bz which has an invariant algebraic surface.

Largest Lyapunov exponents and Lyapunov spectra of desynchronous coupled chaotic maps with finite coupling strength are investigated by considering different map functions. For sufficiently large system size there exists a parameter region where the largest Lyapunov exponents are independent of system size and coupling strength. This parameter region is called the scaling region. Some scaling exponent and scaling function in the distribution of Lyapunov spectra are found. These scaling behaviors are model independent. All the above numerical observations are explained, based on heuristic physical understanding of the competition of intensity of chaoticity, and strength of coupling, and on an analogy of the discrete coupled maps with continuous extended systems.

To detect determinism and instability geometrically, the Lorenz map z_n+1 = f(z_n), which projects a local maximum z_n to the next local maximum z_n+1 of the time series, is useful for such a thin attractor as the Lorenz attractor, but is not applicable for such a "fat" attractor as the Chua circuit. Hence, this study proposes a novel and simple method that is applicable for various attractors. It shows that a T-delayed Lorenz map z_n+T = f_T(z_n), which projects a local maximum z_n to the T-delayed local maximum z_n+T, is more useful. It was found that the appearance of a parabolic image in a T-delayed Lorenz map is a sign of chaos.

A new family of entropy estimators, constructed as a linear combination (weighted average) of nearest neighbor estimators with slightly different individual properties, is proposed. It is shown that a special suboptimal selection of the coefficients in the linear combination results in a polynomial reduction of the estimator's bias, which continues the line of work of P. Grassberger on entropy estimation. Computer simulation results are provided.

Since most prostate tumors are initially hormone-sensitive, hormonal therapy with androgen suppression is a major treatment for them. In this hormonal therapy, however, a tumor relapse is a crucial problem. Androgen-independent tumor cells are considered to be responsible for such a relapse. These cells are not sensitive to androgen suppression but rather apt to proliferate even in an androgen-poor environment. Bruchovsky et al. proposed intermittent androgen suppression (IAS), which may prolong the relapse time when compared with continuous androgen suppression (CAS). IAS therapy is based on switching of medication through monitoring of the serum prostate-specific antigen (PSA). Namely, the medication is suspended when the PSA concentration falls below the lower threshold during on-treatment periods and it is reinstituted when the concentration exceeds the upper threshold during off-treatment periods. In this paper, we propose a model of partial differential equations (PDE) for IAS therapy, on the basis of our previous model of ordinary differential equations, under the assumption that the prostate tumor is a mixed assembly of androgen-dependent (AD) and androgen-independent (AI) cells. Numerical analysis compares the effect of the IAS therapy with that of the CAS therapy for different growth rates of the AI cells, which suggests an optimal protocol of the IAS therapy.

In standard Boolean Networks (BN) the new state of a cell depends upon the neighborhood configuration only at the preceding time step. The effect of implementing memory of different types in cells of BN with different degrees of random rewiring is studied in this article.

We study the thermal excitation of solitons in 1D Toda–Morse lattices in a wide range of temperatures from zero up to physiological level (about 300 K) and their influence on added excess electrons moving on the lattice. The lattice units are treated by classical Langevin equations. The electron distributions are in a first estimate represented by equilibrium adiabatic distributions in the actual fields. Further, the electron dynamics is modeled in the framework of the tight-binding approximation including on-site energy shifts due to electron-lattice coupling and stochastic hopping between the sites. We calculate the electron distributions and discuss the excitations of solectron type (electron-soliton dynamic bound states) and estimate their life times.