International Journal of Bifurcation and Chaos

As Wolfram classification of cellular automata, we have divided a taxonomy of Artificial Micro Worlds (AMWs) into four classes. In previous work, we have presented many examples of self-reproducers belonging to Class 1. In this paper, we present a series of AMWs, belonging to Class 2 (complex AMWs which contain complex species of self-reproducers); Class 3 (which contains species of self-reproducers with chaotic behavior) and Class 4 (which contains species with hyper complex self-reproducers). In particular, we introduce an AMW, belonging to Class 2 (a loop-shaped system); several AMWs, belonging to Class 3 (species with a linear and chaotic behavior) and a series of AMWs, belonging to Class 4 (with species which show very different behaviors). Particular attention will be dedicated to a new type of AMW, which contains a species that has an incorporated developmental program.

Bifurcation analysis has been applied to many power electronics circuits. Literature abounds with results regarding the various ways in which such circuits lose stability under variation of some selected parameters, e.g. via period-doubling bifurcation, Hopf bifurcation, border collision, etc. The current status of research in the identification of bifurcation behavior in power electronics has reached a stage where the salient types of bifurcation behavior, their underlying causes and the theoretical parameters affecting them have been well understood. Currently, the emphasis of research in this field has gradually shifted toward applications that are of direct relevance to practical design of power electronics. One direction is to apply some of the available research results in bifurcation behavior to the design of practical power electronics circuits. The main difficulty is that the abstract mathematical presentations of the available results are not directly applicable to practical design problems. In this paper we will discuss how research efforts may be directed to bridge this gap.

In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N-degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e. periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions.

In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1, 2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature.

We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies E_q of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, …. These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALI_k, k = 2, …, 2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.

With amplitude modulated excitation, the effect on chaotic behavior of Φ⁶-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincaré map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

Soliton-mediated electron pairing (both in real space and in momentum space) is shown to occur in heated one-dimensional (1D) molecular anharmonic systems with Morse interactions.

In this paper, a tri-neuron network with bidirectionally delay and self-feedback is considered. We derive some sufficient conditions dependent or independent of delays for the local stability and instability of this model. Regarding the self-connection delay as the parameter, the Hopf bifurcation analysis is carried out. The direction and stability of the Hopf bifurcation are worked out by applying the normal form theory and the center manifold theory. An example is given and numerical simulations are presented to illustrate the obtained results.

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Duffing equation is a representative nonlinear equation in practical engineering systems, where the most existing research focuses on local solutions of weakly nonlinear systems. In this paper, we study global bifurcations and chaos of the standard Duffing system by employing the Arnold tongue, and use the basin of attraction to investigate the properties of the Arnold tongue overlap. Our results show that a resonance solution and chaos could coexist, when the parameters are on the Arnold tongue overlap. The phenomenon does not exist in a system described by a weak Duffing equation. Numerical solutions for these bifurcations and chaos are also provided to demonstrate the theoretical results.

In this paper, enlightened by the idea of the weight of a polynomial introduced by Swinnerton-Dyer [2002], we find all the invariant algebraic surfaces of the Chen system x′ = a(y - x), y′ = (c - a)x + cy - xz, z′ = xy - bz.

This paper proposes a novel way to look at Chua's circuit and to investigate its chaotic behavior. We propose a systematic method to increase the dimension of Chua's circuit and to choose the element values. Our claims are confirmed by numerical simulations on Chua's circuit and on modified Chua's circuit and also supported by experimental results.

We study the stability of steady states and establish the existence of traveling waves for a diffusive host-vector epidemic with a nonlocal spatiotemporal interaction. We develop the techniques of contracting-convex-sets, limit argument, singular perturbation and fixed point theorems.

We examine the dynamics of laser droplet generation that is dependent on the detachment pulse power. In the absence of the detachment pulse, undulating pendant droplets are formed at the end of a properly fed metal wire due to the impact of the primary laser pulse that induces melting. Eventually, these droplets detach, i.e. overcome the surface tension, because of their increasing mass. We show that this spontaneous dripping is deterministically chaotic by using a positive largest Lyapunov exponent and a negative divergence. In the presence of the detachment pulse, however, the generation of droplets is fastened depending on the pulse power. At high powers, the spontaneity of dripping is completely overshadowed by the impact of the detachment pulse. Still, amplitude chaos can be detected, which similarly as the spontaneous dripping, is characterized by a positive largest Lyapunov exponent and a negative divergence, thus indicating that the observed dynamics is deterministically chaotic with an attractor as solution in the phase space. In the intermediate regime, i.e. for low and medium detachment pulse powers, the two chaotic states compete for supremacy, yielding an intermittent period-doubling to amplitude chaos transition, which we characterize by means of recurrence plots and their properties. Altogether, the transition from spontaneous to triggered laser droplet generation is characterized by a chaos-to-chaos transition with an intermediate dynamically nonstationary phase in-between. Since metal droplets can be used in various industrial applications, we hope that the accurate determination of the dynamical properties underlying their formation will facilitate their use and guide future attempts at mathematical modeling.

Some patterns of synchrony/asynchrony in the dynamics of coupled cell systems can be predicted by symmetry. However, in the system without symmetry, the different patterns of periodic solutions may exist as well. We consider a general model including three cells with multiple time delays that connect in any possible manner. Our approach is based on the analytic construction by using a perturbation procedure together with the Fredholm alternative theory. Then we employ the Poincaré–Lindstedt series expansion to compute the Floquet exponents which determine the stability. Finally, we resort to numerical computation to get some insights about the leading term in the Floquet exponents, and the numerical simulations are given to confirm the theoretical results.

This paper presents some rigorous arguments on a chaotic strange attractor in a spacecraft power system, which is a 3D system with hysteresis switching. By carefully picking a suitable cross-section with respect to the attractor, we find a topological horseshoe of the corresponding third-returned Poincaré map, thus giving a rigorous verification of the existence of chaos in this system. Numerical computation shows that the two Lyapunov exponents of the Poincaré map are both positive and that there exists a two-directional expansion in this horseshoe, suggesting that this attractor is hyperchaotic in nature.

In this paper, source coding or data compression is viewed as a measurement problem. Given a measurement device with fewer states than the observable of a stochastic source, how can one capture their essential information? We propose modeling stochastic sources as piecewise-linear discrete chaotic dynamical systems known as Generalized Luröth Series (GLS) which has its roots in Georg Cantor's work in 1869. These GLS are special maps with the property that their Lyapunov exponent is equal to the Shannon's entropy of the source (up to a constant of proportionality). By successively approximating the source with GLS having fewer states (with the nearest Lyapunov exponent), we derive a binary coding algorithm which turns out to be a rediscovery of Huffman coding, the popular lossless compression algorithm used in the JPEG international standard for still image compression.

An analysis on the nonlinear forced vibration of thermally loaded FGM plate with two simply supported opposite and two free edges subjected to the in-plane and transversal excitations is presented. The material properties of the FGM plates are assumed to be temperature dependent and change continuously throughout the thickness of the plate, according to the volume fraction of the constituent materials based on the power law function. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. The plate is modeled by using the Von Karman hypothesis and the equations of motion are obtained by using an energy approach. It is our aim to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the FGM rectangular plates with two simply supported opposite and two free edges. The equations of motion can be reduced into a two-degree-of-freedom nonlinear system of transverse motion under combined thermal and external excitations by using the Galerkin's method. By the numerical method, the nonlinear dynamical equations are analyzed to find the nonlinear responses of the FGM plate with two simply supported opposite and two free edges. Under certain conditions the periodic motions and the chaotic motions of the FGM plate are found. The bifurcation diagram demonstrates that for a certain geometric and material properties the chaotic responses of the plate exist as the transverse excitation changes. Moreover, numerical simulations also illustrate that the deflections of the nonlinear dynamic of the FGM rectangular plate are larger than that of the periodic motions.

It is well known that second order lowpass interpolative sigma delta modulators (SDMs) may suffer from instability and limit cycle problems when the magnitudes of the input signals are at large and at intermediate levels, respectively. In order to solve these problems, we propose to replace the second order lowpass interpolative SDMs to a specific class of second order bandpass interpolative SDMs with the natural frequencies of the loop filters very close to zero. The global stability property of this class of second order bandpass interpolative SDMs is characterized and some interesting phenomena are discussed. Besides, conditions for the occurrence of limit cycle and fractal behaviors are also derived, so that these unwanted behaviors will not happen or can be avoided. Moreover, it is found that these bandpass SDMs may exhibit irregular and conical-like chaotic patterns on the phase plane. By utilizing these chaotic behaviors, these bandpass SDMs can achieve higher signal-to-noise ratio (SNR) and tonal suppression than those of the original lowpass SDMs.

A lowpass filter inserted into the logistic map stabilizes fixed points of the system for large parameter ranges and increases the range of global stability. With a first order filter we encounter a special case of the general 2-D quadratic map with hyperchaos in parts of its parameter ranges. Filtered maps include some maps with delayed variables, and here we present one of the simplest examples of such a system.

The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.

Marvel et al. [2009] studied sinusoidally coupled phase oscillators, generalizing coupled Josephson junctions. They obtained an explicit reduction of the dynamics to a parametrised family of ODEs on the three-dimensional Möbius group. This differs from the usual reduction on to the orbit space of a symmetry group. We apply the viewpoint of complex projective geometry to obtain an alternative proof that trajectories lie on orbits of the Möbius group, and derive a different explicit form for the reduced ODE. The main innovation is the use of homogeneous coordinates, which linearize the action of the Möbius group and lead to a simple coordinate system in which to write the reduced ODE. We also discuss a Lie-theoretic interpretation.

Let {τ₁, τ₂,…,τ_K} be a collection of nonsingular maps on [0, 1] into [0, 1] and {p₁, p₂,…,p_K} be a collection of position dependent probabilities on [0, 1]. We consider position dependent random maps T = {τ₁,τ₂,…,τ_K;p₁,p₂,…,p_K} such that T preserves an absolutely continuous invariant measure with density f*. A maximum entropy method for approximating f* is developed. We present a proof of convergence of the maximum entropy method for random maps.