International Journal of Bifurcation and Chaos

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

Semi-insulating Gallium Arsenide (SI-GaAs) samples experimentally show, under high electric fields and even at room temperature, negative differential conductivity in N-shaped form (NNDC). Since the most consolidated model for n-GaAs, namely, "the model", proposed by E. Schöll was not capable to generate the NNDC curve for SI-GaAs, in this work we have proposed an alternative model. The model proposed, "the two-valley model" is based on the minimal set of generation-recombination equations for two valleys inside of the conduction band, and an equation for the drift velocity as a function of the applied electric field, that covers the physical properties of the nonlinear electrical conduction of the SI-GaAs system. The "two-valley model" was capable to generate theoretically the NNDC region for the first time, and with that, we were able to build a high resolution parameter-space of the periodicity (PSP) using a Periodicity-Detection (PD) routine. In the parameter-space were observed self-organized periodic structures immersed in chaotic regions. The complex regions are presented in a "shrimp" shape rotated around a focal point, which forms in large-scale a "snail shell" shape, with intricate connections between different "shrimps". The knowledge of detailed information on parameter spaces is crucial to localize wide regions of smooth and continuous chaos.

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

We present the following: (1) if a volume-preserving diffeomorphism has C¹-robustly various limit shadowing property, then it is Anosov; (2) C¹-generically, if a volume-preserving diffeomorphism has various limit shadowing property, then it is Anosov.

This paper investigates the stability and Hopf bifurcation induced by the time delay in a class of (n + 1)-dimension Internet congestion control systems. Although there are several previous works on simplified models of Internet congestion systems with only one or two sources and such works can reflect partly dynamical behaviors of real Internet systems, some complicated problems may inevitably be overlooked. Hence, it is meaningful to study high-dimensional models which stand closer to general realistic large-scale Internet congestion networks. By analyzing the distribution of the associated characteristic roots, we can obtain conditions for keeping systems stable. When the delay increases and exceeds a critical value, the system will undergo a Hopf bifurcation. Furthermore, the explicit formulas to determine the stability and the direction of the bifurcating periodic solution are derived by applying the normal form theory and the center manifold reduction. Finally, two numerical examples are given to verify our theoretical analysis.

This paper presents the bifurcation analysis of fractional order model of delayed single cell which is proposed for delayed cellular neural networks with respect to the time delay τ. The bifurcation points, time delay τ_c, are determined by modified Mikhailov stability criterion for a range of fractional delayed cell order 0.3 ≤ q < 1. Numerical results obtained from Adams–Bashforth–Moulton method demonstrate that the supercritical Hopf bifurcation occurs in the system.

Chaotic systems have been used in random number generation, owing to the property of sensitive dependence on initial conditions and hence the possibility to produce unpredictable signals. Within the types of chaotic systems, those which are defined by only one delay-differential equation are attractive due to their simple model. On the other hand, it is possible to synchronize to the future states of a time-delay chaotic system by anticipating synchronization. Therefore, random number generator (RNG), which employs such a system, might not be immune to the attacks. In this paper, attack on a chaos-based random number generator using anticipating synchronization is investigated. The considered time-delay chaotic system produces binary signals, which can directly be used as a source of RNG. Anticipating synchronization is obtained by incorporating other systems appropriately coupled to the original one. Quantification of synchronization is given by the bit error between the streams produced by the original and coupled systems. It is shown that the bit streams generated by the original system can be anticipated by the coupled systems beforehand.

Uncertain differential system is a type of differential system involving uncertain processes. Stability analysis has been widely studied but no work has been dedicated to attractivity analysis of uncertain differential systems. In this paper, some concepts of attractivity for uncertain differential systems are presented. Then the corresponding sufficient and necessary conditions are given. Furthermore, the stability of the solutions and α-path of uncertain differential systems are studied.

We investigate the saddle-node bifurcation and vibrational resonance in a fractional system that has an asymmetric bistable potential. Due to the asymmetric nature of the potential function, the response and its amplitude closely depend on the potential well where the motion takes place. And consequently for numerical simulations, the initial condition is a key and important factor. To overcome this technical problem, a method is proposed to calculate the bifurcation and response amplitude numerically. The numerical results are in good agreement with the analytical predictions, indicating the validity of the numerical and theoretical analysis. The results show that the fractional-order of the fractional system induces one saddle-node bifurcation, while the asymmetric parameter associated to the asymmetric nature of the potential function induces two saddle-node bifurcations. When the asymmetric parameter vanishes, the saddle-node bifurcation turns into a pitchfork bifurcation. There are three kinds of vibrational resonance existing in the system. The first one is induced by the high-frequency signal. The second one is induced by the fractional-order. The third one is induced by the asymmetric parameter. We believe that the method and the results shown in this paper might be helpful for the analysis of the response problem of nonlinear dynamical systems.

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ_cr. Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

Nonlinear dynamic behaviors of ceramic-metal graded truncated conical shell subjected to complex loads are investigated. The shell is modeled by first-order shear deformation theory. The nonlinear partial differential governing equation in terms of transverse displacements of the FGM truncated conical shell is derived from the Hamilton's principle. Galerkin's method is then utilized to discretize the partial governing equations to a two-degree-of-freedom nonlinear ordinary differential equation. The temperature-dependent materials properties of the constituents are graded in the radial direction in accordance with a power-law distribution. The aerodynamic pressure can be calculated by using the first-order piston theory. The temperature field is assumed to be a steady-state constant-temperature distribution. Bifurcation diagrams, the maximum Lyapunov exponents, wave forms and phase portraits are obtained by numerical simulation to demonstrate the complex nonlinear dynamics response of the FGM truncated conical shell. The influences of the semi-vertex angle, the material gradient index, in-plane and aerodynamic load on the nonlinear dynamics are studied.

Modeling and bifurcation analysis of an energy harvesting system composed of coupled resonators using the Galfenol-based magnetostrictive material are presented. The analysis in this work should be broad enough to be applicable to a large class of vibratory-based energy harvesting systems since various types of vibratory harvesters share the same normal forms, e.g. magnetostrictive and piezoelectric materials. A combined model of the mechanical and electrical domains of a single energy harvester is discussed first. Building on this model, the governing equations of the coupled system are derived, leading to a system of differential equations with an all-to-all coupling between the resonators. A bifurcation analysis of the system equations reveals different patterns of collective oscillations. Among the many different patterns, a synchronous state exists and it is stable over a broad region of parameter space. This pattern has the potential to yield significant increases in power output and it will be used as a starting point to guide future experimental work. A Hamiltonian approach is employed to study analytically the nature of the bifurcations and to calculate an expression for the onset of synchronization valid for any number of harvesters.

In this paper, the chaotic behavior of a driven planar system with two discontinuous terms and a pseudo-equilibrium point in the intersection of the discontinuity surfaces is analyzed. This scenario is not covered by smooth techniques of chaos analysis or other techniques like the extension of Melnikov's method for nonsmooth systems. In consequence, we propose to use an approximate model of the discontinuous system for which this technique can be applied, and compare the responses of both systems, the discontinuous and the approximate, when this last model is close, in a certain way, to the discontinuous system. One of the discontinuous terms, given by a sign function, is approximated by a saturation with high slope at the equilibrium point. Some conditions that determine the chaotic behavior of the approximate system are formally established, and the convergence of its chaotic orbits to some orbits of the discontinuous system, when the slope of the approximation is large enough, is shown. In particular, we show the similarity of the dynamical behavior of both systems where a chaotic behavior can be displayed, for a parameter region determined by the application of the Melnikov technique to nonsmooth systems. A comparison of the Feigenbaum diagrams, for a parameter range obtained from the application of this technique, shows the similarity of their dynamics and the chaotic nature of the discontinuous system.

In this paper, an extended Rikitake system is studied. Several issues, such as Hopf bifurcation, coexistence of stable equilibria and hidden attractor, and dynamics analysis at infinity are investigated either analytically or numerically. Especially, by a simple linear transformation, the wide range of hidden attractors is noticed, and the Lyapunov exponents diagram is given. The obtained results show that the unstable periodic solution generated by Hopf bifurcation leads to the hidden attractor. The existence of hidden attractors that may render the system's behavior unpredictable not only depends on the value of system parameters but also on the value of initial conditions. The phenomena are important and potentially problematic in engineering applications.

A three-dimensional discrete-time Hindmarsh–Rose model obtained by the forward Euler scheme is investigated in this paper. When the integral step size is chosen as a bifurcation parameter, conditions of existence for the fold bifurcation, the flip bifurcation, and the Hopf bifurcation are derived by using the center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Our research results demonstrate that the integral step size makes a difference corresponding to local and global bifurcations of the three-dimensional discrete-time Hindmarsh–Rose model. These results can supply a solid analytical basis to the study of Hindmarsh–Rose model, and it is necessary to illustrate how much the integral step size is adopted in advance when numerical solutions or approximate solutions of the original continuous-time model is concerned.

Actin is a globular protein which forms long filaments in the eukaryotic cytoskeleton, whose roles in cell function include structural support, contractile activity to intracellular signaling. We model actin filaments as two chains of one-dimensional binary-state semi-totalistic automaton arrays to describe hypothetical signaling events therein. Each node of the actin automaton takes state "0" (resting) or "1" (excited) and updates its state in discrete time depending on its neighbor's states. We analyze the complete rule space of actin automata using integral characteristics of space-time configurations generated by these rules and compute state transition rules that support traveling and mobile localizations. Approaches towards selection of the localization supporting rules using the global characteristics are outlined. We find that some properties of actin automata rules may be predicted using Shannon entropy, activity and incoherence of excitation between the polymer chains. We also show that it is possible to infer whether a given rule supports traveling or stationary localizations by looking at ratios of excited neighbors that are essential for generations of the localizations. We conclude by applying biomolecular hypotheses to this model and discuss the significance of our findings in context with cell signaling and emergent behavior in cellular computation.

In this paper, a control strategy based on the inverse optimal control approach is applied for pinning weighted complex networks with chaotic systems at their nodes; additionally, a cost functional is minimized. This control strategy does not require to have the same coupling strength for all node connections.

In this paper, we prove the existence of pullback and uniform attractors for a nonautonomous Liénard equation. The relation among these attractors is also discussed. After that, we consider that the Liénard equation includes forcing terms which belong to a class of functions extending periodic and almost periodic functions recently introduced by Kloeden and Rodrigues [2011]. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate these results with a numerical simulation: we present a simulation showing the pullback attractor for the nonautonomous Van der Pol equation, an important special case of the nonautonomous Liénard equation.

It has recently been shown that the Chen system with c > 0 is identical to the reversed-time Lorenz system with particular negative parameters and that the Chen system with c < 0 is identical to the forward-time Lorenz system with particular negative parameters. This note describes this new regime and shows that it admits chaotic solutions that were previously unexplored in either system.