International Journal of Bifurcation and Chaos

Bifurcational instabilities of increasing complexity are described, with instructive and intriguing applications. Multiple folds in the gravitational collapse of a star triggers a discussion about possible sequencing of folds. Columns, rings and a quirky floating log illustrate safe and dangerous pitchforks, with self-weight models as lecture demonstrations. Localized buckling is employed by biologists using magnetic tweezers to twist a DNA molecule and evaluate a cutting and de-knotting enzyme. Unfolding of complex bifurcations into “structurally-stable” forms by a number (codimension) of perturbations illuminates the imperfection-sensitivity of shells, and the interactive buckling of a guyed mast at a hyperbolic-umbilic catastrophe. Spinning of primitive planets introduces secondary bifurcations, while Kirchhoff’s analogy is displayed for rods and spinning tops. Examples of gyroscopic stabilization and levitation mimic the “traps” that are used in modern experimental physics to hold in place individual electrons and neutrons.

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

A multistable system generated by Piecewise Linear (PWL) subsystems based on the jerk equation is presented. The system’s behavior is characterized by means of the Nearest Integer or the $round(x)$ function to control the switching events and to locate the corresponding equilibria on each of the commutation surfaces. These surfaces are generated through the switching function dividing the space into regions equally distributed along one axis. The trajectory of the system is governed by the eigenspectrum of the coefficient matrix, which can be adjusted by a bifurcation parameter. The behavior of the system can change from multiscroll oscillations in a mono-stable state into the coexistence of several single-scroll attractors in multistable states. The dynamics and bifurcation analysis are illustrated by numerical simulations to depict the multistable states.

We present the development of a new theory of the pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry.

In this paper, the Bogdanov–Takens (B–T) and triple zero bifurcations are investigated for coupled van der Pol–Duffing oscillators with $Z_2$ symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B–T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results.

The popularity of systems of pulse energy conversion (PEC-systems) for practical applications is due to the heightened efficiency of energy conversion processes with comparatively simple realizations. Nevertheless, a PEC-system represents a nonlinear object with a variable structure, and the bifurcation analysis remains the basic tool to describe PEC dynamics evolution. The paper is devoted to the discussion on whether the scientific viewpoint on the natural nonlinear dynamics evolution can be involved in practical applications. We focus on the problems connected with stability boundaries of an operating regime. The results of both small-signal analysis and computational bifurcation analysis are considered in the parametrical space in comparison with the results of the experimental identification of the zonal heterogeneity of the operating process. This allows to propose an adapted stability margin as a sufficiently safe distance before the point after which the operating process begins to lose the stability. Such stability margin can extend the permissible operating domain in the parametrical space at the expense of using cause-and-effect relations in the context of natural regularities of nonlinear dynamics. Reasoning and discussion are based on the experimental and computational results for a synchronous buck converter with a pulse-width modulation. The presented results can be useful, first of all, for PEC-systems with significant variation of equivalent inductance and/or capacity. We believe that the discussion supports a viewpoint by which the contemporary methods of the computational and experimental bifurcation analyses possess both analytical abilities and experimental techniques for promising solutions which could be practice-oriented for PEC-systems.

The past decade has witnessed a boom of wireless communications which necessitate an increasing improvement of data rate, error-rate performance, bandwidth efficiency, and information security. In this work, we propose a quadrature (IQ) differential chaos-shift keying (DCSK) modulation scheme for the application in cognitive radio (CR), named CR-IQ-DCSK, which offers the above improvement. Chaotic signal is generated in frequency domain and then converted into time domain via an inverse Fourier transform. The real and imaginary components of the frequency-based chaotic signal are simultaneously used in in-phase and quadrature branches of an IQ modulator, where each branch conveys two bits by means of a DCSK-based modulation. Schemes and operating principle of the modulator and demodulator are proposed and described. Analytical BER performance for the proposed schemes over a typical multipath Rayleigh fading channel is derived and verified by numerical simulations. Results show that the proposed scheme outperforms DCSK, CDSK and performs better with the increment of the number of channel paths.

We investigate the bifurcation structure of the Kuramoto–Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and $O(2)$ symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov–Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensable for the explanation of the bifurcation structure, the Kuramoto–Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry.

In the present paper, the phenomenon of noise-induced chaos in a piecewise linear system that is excited by Gaussian white noise is investigated. Firstly, the global dynamical behaviors of the deterministic piecewise linear system are investigated numerically in advance by using the generalized cell-mapping digraph (GCMD) method. Then, based on these global properties, the system that is excited by Gaussian white noise is introduced. Then, it is simplified by the stochastic averaging method, through which, a four-dimensional averaged Itô system is finally obtained. In order to reveal the phenomenon of noise-induced chaos quantitatively, MFPT (the mean first-passage time) is selected as the measure. The expression for MFPT is formulated by using the singular perturbation method and then a rather simple representation is obtained via the Laplace approximation, and within which, the concept of quasi-potential is introduced. Furthermore, with the rays method, the MFPT under a certain set of parameters is estimated. However, within the process of analysis, the authors had to face a difficult problem concerning the ill-conditioned matrix, which is the obstacle for the estimation of MFPT, which was then solved by applying one more approximation. Finally, the result is compared with the numerical one that is obtained by the Monte Carlo simulation.

A new chaotic system having variable equilibrium is introduced in this paper. The presence of an infinite number of equilibrium points, a stable equilibrium, and no-equilibrium is observed in the system. Interestingly, this system is classified as a rare system with hidden attractors from the view point of computation. Complex dynamical behavior and a circuital implementation of the new system have been investigated in our work.

We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.

This paper deals with the transient characteristics of the cascade DC–DC Boost converter under a large disturbance by using nonlinear modal series method. Based on the power balance principle, a nonlinear averaged model is derived to describe the nonlinear dynamics of the cascade converter. And then, the modal series method is described in detail and the second order approximate solutions are derived by this method. Furthermore, the simulation results and the theoretical analysis demonstrate that there are plenty of modal interactions, particularly the stronger second order modal interactions, which can affect the transient behavior significantly in the disturbed transient process. The modified approximate solutions considering the dominant nonlinear interaction modes of some state variables are subsequently obtained. In addition, by selecting appropriate system parameters, we can improve the transient behavior of the system under disturbance so that the amplitude and duration of the oscillation can be effectively reduced to satisfy the requirements of the system tolerance during the transient process. Moreover, the dominant oscillation modes of each state variable are also studied, which will help us improve the understanding of the salient transient behaviors in DC–DC cascade converters with disturbance. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

In this paper, we show planar quadratic polynomial differential systems exhibiting as solutions some famous planar invariant algebraic curves. Also we place particular attention to the Darboux integrability of these differential systems.

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

Applying parametric controls to the 3D real-valued Maxwell–Bloch equations, we obtain a Hamilton–Poisson system, a dissipative system with chaotic behavior, and a transitional system between the aforementioned states, which is a conservative system that has only one constant of motion. In the Hamiltonian case, we present some connections of the energy-Casimir mapping with the equilibrium states and the existence of the homoclinic orbits. We study the stability of the equilibrium points of the transitional system and the dissipative system. Furthermore, we point out the chaotic behavior of the introduced system.

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

The Jacobi stability of the normal form of typical bifurcations in one-dimensional dynamical systems is analyzed by introducing the concept of the production process (time-like potential) to KCC theory. This KCC theory approach shows that the geometric invariants of the system characterize the nonequilibrium dynamics of the bifurcations. For example, the deviation curvature that is one of the geometric invariants shows that the well-known two hysteresis jumps in subcritical pitchfork bifurcations differ qualitatively from each other. In the nonequilibrium region, the deviation curvature in the saddle-node and the transcritical bifurcations are a function of the bifurcation parameter alone; thus, the Jacobi stability does not depend on time. However, the deviation curvature in a pitchfork bifurcation is a function of the time-like potential, so the Jacobi stability does depend on time. This time dependence can be described by the Douglas tensor, which is a useful geometric invariant to consider how the higher-order term in the bifurcation system affects the stability structure.

This paper proposes a new approach for constructing higher-dimensional nondegenerate hyperchaotic system with multiple controllers. Here, the so-called higher-dimensional nondegenerate hyperchaotic system means that it can be provided with a maximum number of positive Lyapunov exponents, which has been an open problem for research in recent years. The details of design are given by three steps as follows: (i) Design an $n$-dimensional nominal matrix and similarity transformation matrix, and get an asymptotic stable nominal system; (ii) Add a master controller for the nominal matrix and get the controlled system. Then, find suitable control positions such that the controlled system satisfies the average eigenvalue criterion, i.e. the number of average eigenvalues with positive real parts of all Jacobi matrices over a given period of time is equal to ($n-2$), and the maximum value of average eigenvalues with positive real parts is greater than a given threshold $T_h$; (iii) Add nonmaster controllers, and the control positions are fixed and parameters are given in advance. So it can generate nondegenerate hyperchaotic systems with ($n-2$) positive Lyapunov exponents. Finally, two typical examples are given to show the feasibility and effectiveness of the proposed method.

Circadian clock is an autonomous oscillator which orchestrates the daily rhythms of physiology and behaviors. This study is devoted to explore how a positive feedback loop affects the dynamics of mammalian circadian clock. We simplify an experimentally validated mathematical model in our previous work, to a nonlinear differential equation with two time delays. This simplified mathematical model incorporates the pacemaker of mammalian circadian clock, a negative primary feedback loop, and a critical positive auxiliary feedback loop, $Rev\mathrm{\text{-}}erb\alpha /Cry1$ loop. We perform analytical studies of the system. Delay-dependent conditions for the asymptotic stability of the nontrivial positive steady state of the model are investigated. We also prove the existence of Hopf bifurcation, which leads to self-sustained oscillation of mammalian circadian clock. Our theoretical analyses show that the oscillatory regime is reduced upon the participation of the delayed positive auxiliary loop. However, further simulations reveal that the auxiliary loop can enable the circadian clock gain widely adjustable amplitudes and robust period. Thus, the positive auxiliary feedback loop may provide a trade-off mechanism, to use the small loss in the robustness of oscillation in exchange for adaptable flexibility in mammalian circadian clock. The results obtained from the model may gain new insights into the dynamics of biological oscillators with interlocked feedback loops.

The Sr${}_{0.95}$Ba${}_{0.05}$TiO₃ (SBT) memristor is prepared using the monolayer Sr${}_{0.95}$Ba${}_{0.05}$TiO₃ nano-film structure. In order to apply it into the nonlinear circuit design, the SBT memristor is modeled in the paper. The voltage-controlled physical model of the SBT memristor is established based on its working mechanism. Due to the difficulty in determining the accurate parameters of the voltage-controlled physical model, a flux-controlled mathematical model of the SBT memristor is proposed, and its equivalence relation with the voltage-controlled physical model is proved. Moreover, the parameters of the flux-controlled mathematical model are determined by means of the quadratic polynomial interpolation method using the experimentally measured voltage and current data of the SBT memristor. The simulated $u$–$i$ characteristic curve using the flux-controlled mathematical model coincides well with the measured $u$–$i$ characteristic curves. The result indicates that the flux-controlled mathematical model with the definite parameters can be used to characterize the behaviors of the physical SBT memristor and guide its application to nonlinear circuit design.