International Journal of Bifurcation and Chaos

Although initial results that contributed to the emergence of the nonlinear dynamical system theory arose from astronomy (the three-body problem), many subsequent developments were related to radio engineering and electronics. The path between the van der Pol equation and the Chua circuit is thus reviewed through main historical contributions.

In this paper, periodic and chaotic behaviors in the Chua circuit system are investigated, and the analytical prediction of periodic flows in such a system is carried out. The solutions of the system in different regions with different parameters are first obtained. The switching boundaries are introduced for systems switching because of different system parameters in different domains. In the vicinity of the switching boundaries, the normal vector-field product is introduced to measure flow switching on the separation boundary, and the conditions for grazing and passable flows to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from mapping structures. The local stability and bifurcation analysis are carried out.

Nested invariant 3-tori surrounding a torus braid of elliptic type are found to exist in a model of a fluid flow with quasiperiodic forcing. The Hamiltonian describing the system is given by the superposition of two steady stream functions, one with an elliptic fixed point and the other with a coincident hyperbolic fixed point. The superposition, modulated by two incommensurate frequencies, yields an elliptic torus braid at the location of the fixed point. The system is suspended in a four-dimensional phase space (two space and two phase directions). To analyze this system we define two three-dimensional, global, Poincaré sections of the flow. The coherent structures (cross-sections of nested 2 tori) are found each to have a fractal dimensional of two, in each Poincaré cross-section. This framework has applications to tidal and other mixing problems of geophysical interest.

In this paper, we propose a qualitative and quantitative dynamical study about the evolution of nonsmooth torus in a Digital Delayed Pulse-Width Modulator (PWM) switched buck converter. We explain the birth and destruction of the torus by successive discontinuity induced bifurcations (DIBs). The Digital-PWM control is based on Zero Average Dynamics (ZAD) strategy and a one-period delay is included in the control law. The control parameter (k_s) of the ZAD strategy can be varied in a large range, ideally (-∞, ∞). On these borders, the dynamical behavior is the same and thus an annulus-like parameter space is considered. Under variation of k_s, the system gets closer to a codimension-two bifurcation point, where two simultaneous border-collision bifurcations (BC) meet. The system leaves the high-order periodic behavior and a high-order band torus appears. Wrinkles and nonsmoothness due to more and more successive border-collisions bifurcations cause the torus destruction in high-order band chaos with tent-map-like structures and Mandelbrot-like sets. Finally, the chaotic bands merge in one-band chaos. The switched converter is modeled as a piecewise linear system where an analytical expression of the Poincaré map is available. Characteristics as duality, symmetry and recurrent behavior are determined using additional numerical methods for the Poincaré map decomposition in periodic sequences.

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary "critical" region containing a smooth curve of tangencies between certain canonical "stable" foliations.

It is well known that the stability of a homoclinic loop for planar vector fields is closely related to the cyclicity of this homoclinic loop. For a planar homoclinic loop consisting of a hyperbolic saddle, the loop values are crucial to the stability. The loop values are divided into two classes: saddle values and separatrix values. The saddle values are related to Dulac map near the saddle, and the separatrix values are related to the regular map near the homoclinic loop. The alternation of these quantities determines the stability of the homoclinic loop. So, it is important to investigate the separatrix values in both theory and for practical applications.

For a given planar vector field, we can try to calculate the saddle values by means of dual Liapunov constants or by finding elementary invariants developed by Liu and Li [1990]. The first separatrix value was obtained by Dulac. The second separatrix value was given by Han and Zhu [2007] and by Hu and Feng [2001] independently. The third separatrix value was obtained by Luo and Li [2005] by means of Tkachev's method. In this paper, we shall establish the formulae for the third and fourth separatrix values. As applications, we will give an example with the homoclinic bifurcation of order 9 and prove that the cyclicity of homoclinic loop together with double homoclinic loops is 57.

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

In this paper, we consider two nonlinear equations u_t - u_xxt + 3u²u_x = 2u_xu_xx + uu_xxx and u_t - u_xxt + 4u²u_x = 3u_xu_xx + uu_xxx which are called modified forms of Camassa–Holm and Degaperis–Procesi equations, respectively. For given constant wave speed, we investigate the coexistence of multifarious explicit nonlinear wave solutions, solitary wave solution, peakon wave solution, singular wave solution, smooth periodic wave solution and singular periodic wave solution. Not only is the coexistence shown, but the concrete expressions are presented via the bifurcation method of dynamical systems. From our work it can be seen that these two equations possess a lot of similar properties, and the types of their explicit nonlinear wave solutions are more than that of Camassa–Holm and Degaperis–Procesi equations. Many previous results are our special cases.

This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

A method of identifying the transition rule, encapsulated in a modified cellular automata (CA) model, is demonstrated using experimentally observed data of the evolution of dendritic crystal growth patterns in NH₄Br crystals. The influence of the factors, such as experimental set-up and image preprocessing, color and size calibrations, on the method of identification are discussed in detail. A noise reduction parameter and the diffusion velocity of the crystal boundary are also considered. The results show that the proposed method can in principle provide a good representation of the dendritic growth anisotropy.

We study the structure of point clouds obtained as time delay embeddings of human speech signals by approximating the data sets with certain simplicial complexes and analyzing their persistent homology. Results for several different sounds are presented in embedding dimensions 3 and 4. The first Betti number allows a coarse classification of sounds into three groups: vowels, nasals and noise.

In the present paper, the smoothness of a Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS), as described by its Lipschitz exponent, is investigated. This is achieved by considering the simulation of a generally uneven surface using CHFIS. The influence of free variables and Lipschitz exponent on the smoothness of CHFIS is demonstrated by considering interpolation data generated from a sample surface.

David Bohm had shown that the Schrödinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions — action and probability density. The first equation is the Hamilton–Jacobi (HJ) equation, a "visiting card" of classical mechanics, is modified by the Bohmian quantum potential. This potential is a nonlinear function of the probability density. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed into two Bohmian quantum correctors. The first corrector modifies the kinetic energy term of the HJ equation, and the second one modifies the potential energy term. The unification of the quantum HJ equation and the entropy balance equation gives a complexified HJ equation containing complex kinetic and potential terms. The imaginary parts of these terms have an order of smallness about the Planck constant. The Bohmian quantum corrector is an indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to an imaginary sector. The difference between the Bohmian and Feynman's trajectories is that the former satisfies the principle of least action and they bifurcate on interfaces. The latter covers all possible paths from a source to a detector. They can split and annihilate.

We present the results of a very simple nonlinear nonautonomous parallel LCR circuit which exhibits a rich variety of bifurcations and chaos. This simple experimental chaotic oscillator is made up of a few passive components and a uniquely designed nonlinear element comprising of a junction diode and Op-amp based negative conductor. The system exhibits a rich dynamics such as chaos via torus breakdown, period-doubling, period three-doubling, period-adding, Farey sequences, etc., and the results of the investigation are vividly presented along with experimental, numerical and analytical evidences. The laboratory experimental outcomes are in agreement with the numerical and analytical results.

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

Certain systems present chaotic dynamics when subjected to a regular periodic input. In a study of a nonlinear model of an electromechanical transducer, its dynamic stability is analyzed and it is observed to present chaotic dynamics when a squared signal is introduced as input to the excitor circuit voltage. It is demonstrated that the chaotic movement is due to the periodic modification in the attraction basin of the state space, caused by the input varying in time. Varying the input causes the system to cross saddle type bifurcation values in which points of equilibrium appear and disappear, periodically modifying the qualitative aspects of the system's phase space. This paper describes the deterministic chaos generation by the regular and periodic modification of the properties of the phase space.

In this paper, we investigate the intertwined basins of attraction. Using a general definition, we discuss the local intertwining properties and present an easy condition to guarantee the existence of intertwined basins for planar dynamical systems.

We investigate properties and examples of iterated planar piecewise parabolic (PWP) maps, including those induced by two-torus parabolic maps and their inverses. PWP maps are area-preserving maps that have constant linearizations but only one eigenvector with eigenvalue one. We obtain necessary and sufficient conditions for a two-torus parabolic map to be invertible.

For noninvertible PWP maps, there are a number of questions about the existence and structure of attractors for such maps. We introduce the simplest example of a nontrivial PWP map, defined by a parabolic linear map with a translation that is a different constant on each side of a given line in the plane. For this "half plane parabolic map" we obtain sufficient conditions for a bounded attractor to exist, discuss their dynamical properties and give some examples of these and related maps.

The counter-intuitive appearance of stable periodic orbits in three-dimensional piecewise linear systems has been recently reported for stable saturated control systems with high gain and real triple eigenvalues [Moreno & Suárez 2004]. In this letter using several mathematical tools, the reported sudden phenomenon is explained, and the analysis is completed by providing a global bifurcation diagram of the symmetric periodic orbits.

Approximate methods are used only to illustrate the nonlinear behavior with respect to the bifurcation parameter. Analytical methods are employed to rigorously prove the main result. The employed techniques are useful not only for the family studied but also for generic three-dimensional symmetric piecewise linear systems.

The Hopf cyclicity of nonsmooth Liénard systems on the plane is studied and an algebraic method to find the Hopf cyclicity is presented. A sufficient and necessary condition which ensures the origin being a center is obtained. Some new and interesting applications are presented.

We study resonant bifurcations of periodic orbits of six-dimensional systems associated with three pairs of conjugate pure imaginary eigenvalues. Bifurcation functions can be obtained by Liapunov–Schmidt methods and we can use these functions for a detailed analysis of six-dimensional systems about their resonant bifurcations of periodic orbits.

We address a nonautonomous differential equation with a pulse function, whose moments of discontinuity depend on the initial moment. Existence of a quasi-minimal set is proved. An appropriate simulation of a chaotic attractor is presented.

This paper addresses the cryptanalysis of three chaos-based secure communication schemes recently proposed by Yu et al. [2007]. Attacks with adaptive observers designed by the dynamical minimization algorithm are assumed. Considering Kerckhoff's principle, i.e. assuming that the basic structure of a cryptosystem is known but some user-specified parameters remain in secret, it is demonstrated that the covered messages in all these schemes can be estimated and hence their securities are challenged.

Diabetic neuropathy may be a result of disturbed vasomotion, the spontaneous rhythmic oscillations of vessel diameter, flow, or pressure in the microcirculation. To examine the differences in vasomotion prior to overt diabetes and with diagnosed diabetes, metabolic and blood flow measurements were obtained from three metabolically distinct groups of nonhuman primates: normoglycemic, metabolic syndrome/prediabetic, and overtly diabetic. Three of these indices characterize spectral properties of microcirculatory vasomotion: (1) the fraction of spectral power in a band around 2.0 Hz, (2) the ratio of spectral power in a band around 1.0 Hz to that around 2.0 Hz, (3) the Hurst exponent. The rest were more traditional measures including (4) fasting plasma Glucose (FPG), (5) glycosylated hemoglobin level (HbA1c), (6) fasting insulin levels (IRI), (7) average systolic blood pressure, and (8) average diastolic blood pressure.

We found that the mean fraction of the spectral power of the vasomotion in the 2 Hz band is significantly greater in the nondiabetic than in the diabetic group. Similarly, the ratio of the spectral power in the 1.0 Hz band to that in the 2.0 Hz band was significantly higher among the diabetic subjects than in the nondiabetic and prediabetic subjects.

Principal component analysis (PCA) of the ensemble of eight measures revealed that values of the first principal component in the diabetic group were highly and significantly different from those of the normal and prediabetic groups, which are also significantly different from each other. Furthermore, the spectral measures of vasomotion offered a distinct advantage over the traditional physiological variables in distinguishing the normal subjects from the prediabetic group, and are likely to reflect very early autonomic dysfunction.

The synchronization of metacommunities due to dispersal among patches is analyzed in the case of slow–fast populations. The analysis is performed by studying a standard model with the fast population dispersing when special meteorological conditions are present. This assumption fits very well with the peculiar nature of slow–fast systems and implies that metacommunities synchronize if the slow population accelerates during the outbreak of the fast population. This result shows great potential in the study of marine and fresh-water plankton communities as well as in the study of synchronization of insect-pest outbreaks in forests.

In this paper, a class of fuzzy cellular neural networks (FCNNs) with distributed delays and time-varying coefficients is studied. By employing the mathematical analysis method, some sufficient conditions are derived for ensuring that all solutions of such FCNNs converge exponentially to zero. Both the Lipschitz continuous condition on activation functions and the differentiability of time-varying delays are not required in this study.