International Journal of Bifurcation and Chaos

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented.

In this paper we study the existence of traveling waves with any rational velocity in coupled circle maps. We introduce an induced map, then a traveling wave with velocity p/q corresponds to a fixed point of the induced map. Moreover, the stability of a traveling wave is equivalent to that of the corresponding fixed point. Space translational system is chaotic on the set of traveling waves with rational velocities. In addition, time evolution operator exhibits sensitivity-like behavior with respect to initial conditions. By investigating the spatial periodicity of traveling waves, we obtain infinitely many space-time periodic structures. We also consider spatial asymptoticity of these traveling waves, which leads to the existence of fronts, defect solutions and soliton-like solutions. The abundance of traveling waves may be regarded as a signature of the spatial-temporal complexity in extended systems.

The classical dynamics of the triaxial ellipsoidal billiard with isotropic harmonic potential attracting to the center of the ellipsoid is discussed. The integrability preserving potential introduces an energy dependence to the foliation of energy shells into invariant tori. This foliation and the character of the corresponding motion is described in terms of 13 qualitatively different energy surfaces in the space of the action variables. Frequencies and the location of resonances are calculated. The consequences of the superintegrability of the low-energy case, the isotropic harmonic oscillator, for the energy surfaces in action space are investigated.

In this paper we compare the disruption of a second-order type II phase-locked loop (PLL) by two different waveforms: a sinusoid and a sawtooth. The choice of these two waveforms results from a novel approach to the problem of determining an appropriate forcing function for studying the disruption of such systems. It is shown that the sawtooth is the better disruptor for low frequencies of modulation, i.e. in the regime of physical interest. Analytical, numerical and experimental results are reported which support this conclusion. The paper presents the first results of a sawtooth modulated PLL, which viewed as a nonsmooth dynamical system with a nonsmooth forcing function is a problem of considerable theoretical interest. This paper demonstrates that forcing functions can be designed for nonlinear dynamical systems which are well suited to the study of certain aspects of the system dynamics, and significantly better than the conventional choice of a simple sinusoid.

The paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is two-dimensional and the Smale horseshoe is constructed. Moreover, we also illustrate the variant of CNN, closely related to Henón-type and Belykh maps, and discrete Allen–Cahn equations.

This work investigates the global mosaic pattern and spatial entropy for one-dimensional cellular neural network (CNN). A novel method is developed to partition the parameter space into finitely many regions. The CNNs, with parameters in each region, have the same global pattern. An algorithm is also presented to evaluate the spatial entropy.

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called "figure-of-eight," and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. The ensuing resonant gluing bifurcations exhibit features of both gluing bifurcations and resonant homoclinic bifurcations. Under certain twist conditions, the bifurcation structure is extremely rich, although describing zero-entropy flows. The analysis carefully exploits the topology of the orbits, the centre manifold and the parameter space.

Criteria for occurrence of transient tumbling chaos in the parametrically driven pendulum are examined by computer aided methods. It is shown that stable manifolds of the saddles associated with the rotating attractors play a crucial role in separating the oscillating and rotating attractors in the phase-plane. A sequence of global bifurcations related to the saddles generates the fractal structure of the basins of attraction of the coexisting attractors. The fractal structure of the basin boundaries implies occurrence of the transient tumbling motion, i.e. the transients that involve rotations with changing directions and oscillatory motion.

A technique for constructing the transformations to real Hamiltonian normal forms of linear Hamiltonian systems via permutation matrices is presented. In particular, a method is shown for obtaining the symplectomorphism between the symplectic basis of the real Jordan form to the standard symplectic basis in which the real Hamiltonian normal form resides. All possible degeneracies are accounted for since the algebraic and geometric multiplicities of nonsemisimple eigenvalues are not restricted, including the "difficult" cases of zero and imaginary eigenvalues. Since the normal forms are not unique, several possible arrangements of the suggested transformations are given which result in the various normal forms derived previously as well as in a few new ones for degenerate cases which have not appeared before.

The general design for dead-beat and asymptotic synchronizers for a large class of discrete-time chaotic systems is proposed. According to whether the form of the transmitter output (drive signal) is linear, nonlinear or the sum of two, different system structures for synchronization discussions are held. Secure communications is then applied taking into consideration to which state in the transmitter masks the message. Examples of different secure communication schemes are discussed, with a comparison given of the various schemes based on the performance of the receivers ability to recover the message. To accomodate the uncertainty existing in the transmitter parameters, an extended Kalman filter (EKF) algorithm is utilized to estimate both the parameters and states when the message is already embedded. To overcome the problem of high error rates of recovered messages while simultaneously estimating parameters, two alternative methods, namely linear output scheme and indirect scheme, are presented to improve the performance. Numerical simulations for secure communications illustrate a binary signal as the message is recovered and recognizable at the receiver's end.

Using a model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce a feedback control strategy of chemotherapy in an early treatment setting, where the control represents the percentage of effect chemotherapy has on the viral production. We seek to regulate the viral count by manipulating the percentage of effect chemotherapy has on the viral production. We show via numerical simulations that the proposed feedback control strategy can handle strong uncertainties in the HIV dynamics induced by imperfect modeling and sampled/delayed cell measurements.

In this paper, we develop and apply some digital design and redesign techniques for ordering the chaotic Chua's circuit. The idea of using sampled-data feedback for controlling the circuit was previously suggested [Yang & Chua, 1998], which relies on small sampling periods. We show how this sampled-data feedback control method can be significantly improved, so that large sampling times are allowed, for the same purpose of ordering the nonlinear circuit, from anywhere within the chaotic attractor towards a predesired periodic cycle of the circuit.

Experimental results from a classical Faraday experiment have been obtained using a point-localized measurement of the displacement of the fluid surface. We used a capacitive transducer as probe, and have been able to study the period-doubling bifurcations in time domain through the determination of cross-correlation and power spectrum.

The presence of deterministic chaos has been clarified experimentally in the time series of pulsation of capillary vessels [Tsuda et al., 1992]. In this paper, first we reconfirmed the nonlinearity of the time series of pulsation of capillary vessels with surrogate data testing. Second, we investigated the characteristics of chaos of capillary data on normal subjects in different states of the autonomic nervous system and on psychiatric patients. As a result, the chaotic fluctuations decreased for the state of sympathetic dominance of normal subjects. On the contrary, the chaotic fluctuations increased for the state of parasympathetic predominance of normal subjects. We also found that the chaotic fluctuations decreased in the symptomatic phase of psychiatric patients. Interpretation of this fact is as follows.

Psychiatric patients in symptomatic phase are in a condition of poor flexibility of emotion and motivated behavior in the limbic system. This condition affects the regulation of the autonomic nervous system in the hypothalamus. So it is easy to fall into the imbalance of the autonomic nervous system even with slight stress. Furthermore, in the attractor of capillary chaos we found two parts that showed the role of origin of stimulation of the sympathetic nervous system and that of stimulation of the parasympathetic nervous system, respectively. The geometrical change of topology was caused by the interplay of two parts. From our study we concluded that chaos of pulsation in capillary vessels reflected mental conditions through the autonomic nervous system.

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

To generate strange nonchaotic attractor in quasiperiodically driven systems, there must be an unstable region in its phase-space. In this paper, a theoretical analysis shows that the quasiperiodic force acts as noise to lead the trajectory running into different expanding orbits when the trajectory repeatedly runs into the unstable region. Thus the resulting attractor is strange. The local-phase Lyapunov exponent is introduced for the study of low-frequency quasiperiodically driven systems. It is shown that the local-phase Lyapunov exponents can be approximated by the exponents of autonomous systems. The statistical properties of SNA system driven by low-frequency quasiperiodic force can then be approached by a set of autonomous systems.

A fractal can be generated by using recursion, and a fractal set of data can be analyzed in the same way. This paper discusses a box-counting algorithm which exploits self-similarity via recursion. The speed of the algorithm is O(n), where n is the number of points in the set. The algorithm is applied to multifractal analysis of the binomial multiplicative process. The algorithm does not involve sorting the floating-point summation; a brief examination of the sorting issue is presented. Scattering for negative q, characteristic of box-counting algorithms, is encountered, and is explained for the simple case of a uniform 1-D set.