International Journal of Bifurcation and Chaos

In this tutorial the chaotic behavior of the fractional-order Chua's circuit is investigated from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which enables the solution of the fractional differential equations to be found in closed form. By exploiting the capabilities offered by the decomposition method, the paper presents two remarkable findings. The first result is that a novel bifurcation parameter is identified, that is, the fractional-order q of the derivative. The second result is that chaos exists in the fractional Chua's circuit with order q = 1.05, which is the lowest order reported in literature for such circuits. Finally, a reliable and efficient binary test for chaos (called "0–1 test") is utilized to detect the presence of chaotic attractors in the system dynamics.

The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamard's graph transformation.

We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval.

We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

We discuss a general mechanism that induces periodic switching between different dynamic phenomena in rings of cells. Such intermittent behavior is normally modeled by asymptotically stable heteroclinic cycles. Our mechanism does not require a heteroclinic cycle in the ring of cells, instead a Van der Pol oscillator forces our system so that it periodically changes state.

We investigate this mechanism numerically in a number of different examples. These simulations demonstrate a variety of interesting forms of intermittency.

In this paper, the dynamics of discrete-time RTD-based cellular neural networks is studied. By virtue of topological horseshoe theory, it can be shown that the dynamics of the map derived from the discretization of RTD-based cellular neural networks is semi-conjugate to the dynamics of three-shift map.

We study the asymptotic behavior of nonautonomous discrete Reaction–Diffusion systems defined on multidimensional infinite lattices. We show that the nonautonomous systems possess uniform attractors which attract all solutions uniformly with respect to the translations of external terms when time goes to infinity. These attractors are compact subsets of weighted spaces, and contain all bounded solutions of the system. The upper semicontinuity of the uniform attractors is established when an infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. We also examine the limiting behavior of lattice systems with almost periodic, rapidly oscillating external terms in weighted spaces. In this case, it is proved that the uniform global attractors of nonautonomous systems converge to the global attractor of an averaged autonomous system.

This paper is concerned with chaotification for a class of first-order partial difference equations, in which the system size is finite or infinite. Nine new chaotification schemes for the class of first-order partial difference equations with general controllers, mod-operation, and sawtooth functions are established, respectively. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. In addition, five new chaotification schemes for general discrete dynamical systems in finite-dimensional real spaces and l^∞ are established. Two illustrative examples are provided with computer simulations.

The convergent dynamical behaviors of a class of impulsive integro-differential equation are discussed. By establishing an integro-differential inequality with impulsive initial conditions and using the properties of M-cone and eigenspace of the spectral radius of non-negative matrices, some new sufficient conditions to guarantee the global exponential stability are obtained. The results extend and improve the earlier publications. An example is given to illustrate the theoretical results.

A parameter estimation method for chaotic systems with low-dimensional chaos is proposed. Such systems display peak-to-peak dynamics, namely their chaotic dynamics can approximately (but accurately) be described by means of a 1-D map, which is directly derivable from a time series of the system output. The method essentially consists of searching for the best matching between the 1-D map generated with a parameter-dependent, a priori defined continuous-time model, and the 1-D map derived from the data. The application of the method to several examples proves that it is quite accurate and fairly insensitive to the measurement noise.

The bifurcations of multiple limit cycles for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The governing nonlinear equation of motion is established for the rotor-AMB system with single-degree-of-freedom and parametric excitation. Using the method of multiple scales, the governing nonlinear equation of motion is first transformed to the averaged equation, which is in the form of a Z₂-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, the bifurcation theory of planar dynamical system and the method of detection function are utilized to analyze the bifurcations of multiple limit cycles of the averaged equation. Four groups of parametric controlling conditions are given to obtain the configurations of compound eyes. It is found that there exist respectively at least 17, 19, 21 and 22 limit cycles in the rotor-AMB system with the time-varying stiffness under the different controlling conditions.

This paper provides further results on bifurcation analysis of indirect field oriented control of induction motors. Previous results presented on this subject [Bazanella & Reginatto, 2000, 2001; Gordillo et al., 2002] are summarized and extended by means of a codimension-two bifurcation analysis. It is shown that codimension-two bifurcation phenomena, such as a Bogdanov–Takens and zero–Hopf bifurcations, occur in IFOC as a result of parameter mismatch and certain setting of the proportional-integral speed controller. Conditions for the existence of such bifurcations are derived analytically, as long as possible, and bifurcation diagrams are presented with the help of simulation and numerical bifurcation analysis.

In this paper we present a computer-assisted technique that allows us to prove rigorously that a transversal homoclinic orbit of discrete dynamical systems with this method is simpler than previous works on this subject.

Further reduction for classical normal forms of smooth maps is considered in this paper. Firstly, based on the idea of computation of simplest normal forms for vector fields [Yu & Yuan, 2003], we compute the transformed map of a given smooth map under a near identity formal transformation, and give a recursive formula for the homogeneous terms of the transformed map, which is a powerful tool for further reduction of classical normal forms of smooth maps. Secondly, by using the recursive formula, the idea of [Chen & Della Dora, 1999] for further reduction of normal forms for maps and the method introduced by Kokubu et al. [1996] for further reduction of normal forms of vector fields, we develop the concepts of Nth order normal forms and infinite order normal forms of smooth maps, and give some sufficient conditions for uniqueness of normal forms of smooth maps. As an application, we show the occurrence of the flip–Neimark–Sacker bifurcation in a financial model.

An algebraic approach is proposed for the fast, accurate, identification of the unknown parameters in Chua's chaotic oscillator. The proposed algorithm uses the availability of two measurable output voltage signals and produces an exact formula for the unknown parameters, which may be realized in terms of iterated convolutions. We show that Chua's system parameters are linearly identifiable, with respect to the two proposed measurable outputs, thus allowing us to obtain a linear system for the unknown parameters from where these unknowns are readily obtained. Suitable algebraic operations on the output differential equations make the proposed algorithm independent of the unavailable initial conditions of the underlying nonlinear dynamical system and robust with respect to high frequency output measurement noises. Suitable algorithm reinitialization, or resetting of the integrations, allow for the efficient computation of piecewise constant varying parameters. Convincing computer simulations are presented and discussed.

This letter proposes a procedure for generating strongly chaotic beats that have been hardly obtainable hitherto. The beats are produced in a nonlinear optical system governing second-harmonic generation of light. The proposition is based on the concept of an optical coupler, but can be easily adopted to other nonlinear systems and Chua's circuits.

A system for generating a multi-butterfly chaotic attractor using the multi-level-logic pulse-excitation technique is proposed. Two-butterfly, three-butterfly and four-butterfly attractors are demonstrated. Results from an experimental setup are also shown.

We study synchronization processes in networks of slightly nonidentical chaotic systems, for which a complete invariant synchronization manifold does not rigorously exist. We show and quantify how a slightly dispersed distribution in parameters can be properly modeled by a noise term affecting the stability of the synchronous invariant solution emerging for identical systems when the parameter is set at the mean value of the original distribution.

Random number generators are widely used in different applications. However, it is difficult to obtain a good random number generator in low precision and resource constrained system, such as an eight-bit micro-controller system which is still commonly used in industrial and consumer markets. This paper provides a practical solution for this problem based on chaotic systems. By the use of a modified Chua's circuit, it is demonstrated that the sampled state, after post-processing by a high-dimensional chaotic map, can be used as a random source even in an eight-bit environment. The randomness of the generated sequence is testified and confirmed by different statistical tests and the up-to-date statistical suite.

We study the transition from stochasticity to determinism in the three-strategy pair-approximated prisoner's dilemma game. We show that the stochastic solution converges to the deterministic limit cycle attractor as the number of participating players increases. Importantly though, between the stochastic and periodic solutions, we reveal a broad range of population sizes for which the system exhibits deterministic behavior, yet fails to settle onto the limit cycle attractor. We show that these states are characterized by chaos via a rigorous treatment. Results are discussed in view of their sociological importance.

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained.

We study electron-electron pairing in an one-dimensional model lattice system embedded into a three-dimensional environment. The electron pair potential is lowered by a single, localized lattice deformation. Such a deformation is related to solitons moving along the lattice. Yet the exact form and the time evolution of the lattice excitation are of secondary relevance as the electron pair is stable for sufficiently wide deformations which propagate on molecular time scales, e.g. velocity of sound ≪ electron velocity. The spatial structure of the pair potential and the electron-electron wave function bring a mechanism of pairing different from the exchange of phonons between the electrons and the lattice which leads to Cooper pairs, and different also from the formation of bipolarons.

The standard model (SM) for high-energy physics describes fundamental interactions between subatomic particles down to a distance scale on the order of 10^-18 m. Despite its widespread acceptance, SM operates with a large number of arbitrary parameters whose physical origin is presently unknown. Our work suggests that the generation structure of at least some SM parameters stems from the chaotic regime of renormalization group flow. Invoking the universal route to chaos in systems of nonlinear differential equations, we argue that the hierarchical pattern of parameters amounts to a series of scaling ratios depending on the Feigenbaum constant. Leading order predictions are shown to agree reasonably well with experimental data.