International Journal of Bifurcation and Chaos

This paper is primarily a study of special families of rational maps of the real plane of the form:

Using the damped harmonic oscillator equations as a mathematical template, several novel chaotic oscillators are developed with an emphasis on mathematical simplicity and ease of electronic circuit implementation. These chaotic systems offer an intuitive introduction to chaos theory, enabling comparison of mathematical and computational analyses with experimental results.

A mathematical model is proposed for interpreting the love story portrayed by Walt Disney in the film "Beauty and The Beast". The analysis shows that the story is characterized by a sudden explosion of sentimental involvements, revealed by the existence of a saddle-node bifurcation in the model. The paper is interesting not only because it deals for the first time with catastrophic bifurcations in specific romantic relationships, but also because it enriches the list of examples in which love stories are satisfactorily described through Ordinary Differential Equations.

A predator–prey system with neutral delay is investigated from the viewpoint of bifurcation analysis on neutral delay differential equations. Stability and Hopf bifurcation of the inner equilibrium are given, by which we show how the neutral terms affect the dynamical behavior of the prey and the predator. To give more detailed information on the periodic oscillations, the direction and stability of Hopf bifurcation are studied by using the normal form theory of neutral equation. We find neutral delay makes the predator–prey system more complicated and usually induces stability switches or double Hopf bifurcations. Near the double Hopf bifurcation we give the detailed bifurcation set by calculating the universal unfoldings. It is shown that the population of prey or predator may exhibit transient quasiperiodic oscillations driven by the neutral delay. Finally, we carry out several groups of illustrations.

In this paper, chaos and adaptive synchronization for a fractional-order Genesio–Tesi system with fifth order nonlinearity are investigated. The minimum effective dimension for the system to remain chaotic is 2.784 in the commensurate-order case and 2.793 in the incommensurate-order case. A period-doubling bifurcation and an interior crisis from single-scroll to double-scroll attractors are also found with the variation of derivative order. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization of the system with unknown parameters is realized by designing appropriated controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.

The aim of this paper is to address the global asymptotic stabilization (GAS) of the class of affine control systems which are open-loop point-dissipatives (have compact global attractors), that include some chaotic systems (e.g. Lorenz, Chen, smooth Chua), via admissible (bounded and regular) feedback controls. The existence of the global attractor for the free dynamics, whose overestimation is achieved by a Lyapunov function V(x), allows us to use V(x) as a basis to obtain a control Lyapunov function (CLF) in order to apply our control design method (entailed from Artstein and Sontag's CLF-stabilization approach) developed for the GAS of affine control systems via admissible feedback controls. We illustrate the results with the chaos control of a smooth (cubic) Chua's system via admissible feedback controls.

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

This paper investigates the complete synchronization of two Boolean networks via logic control. Both feedback control and open-loop control are proposed to make the slave network completely synchronized with the master Boolean network. Using the algebraic state-space representation of Boolean networks, we derive several necessary and sufficient conditions for complete synchronization between two Boolean networks. Two examples are given to illustrate the obtained results.

This work discusses the applicability of a method for phase determination of scalar time series from nonlinear systems. We apply the method to detect phase synchronization in different scenarios, and use the phase diffusion coefficient, the Lyapunov spectrum, and the similarity function to characterize synchronization transition in nonidentical coupled Rössler oscillators, both in coherent and non-coherent regimes. We also apply the method to detect phase synchronous regimes in systems with multiple scroll attractors as well as in experimental time series from coupled Chua circuits. The method is of easy implementation, requires no attractor reconstruction, and is particularly convenient in the case of experimental setups with a single time series data output.

In this paper, we investigate the complex dynamical behaviors of a biological network that is derived from innate immune responses and that couples positive and negative feedback loops. The stability conditions of the non-negative equilibrium points (EPs) of the system are obtained, using the theory of dynamical systems, and we deduce that no more than three stable EPs exist in this system. Through bifurcation analysis and numerical simulations, we find that the system presents rich dynamical behaviors, such as monostability, bistability and oscillations. These results reveal how positive and negative feedback cooperatively regulate the dynamical behavior of the system.

Music is an amalgam of logic and emotion, order and dissonance, along with many combinations of contradicting notions which allude to deterministic chaos. Therefore, it comes as no surprise that several research works have examined the utilization of dynamical systems for symbolic music composition. The main motivation of the paper at hand is the analysis of the tonal composition potentialities of several discrete dynamical systems, in comparison to genuine human compositions. Therefore, a set of human musical compositions is utilized to provide "compositional guidelines" to several dynamical systems, the parameters of which are properly adjusted through evolutionary computation. This procedure exposes the extent to which a system is capable of composing tonal sequences that resemble human composition. In parallel, a time series analysis on the genuine compositions is performed, which firstly provides an overview of their dynamical characteristics and secondly, allows a comparative analysis with the dynamics of the artificial compositions. The results expose the tonal composition capabilities of the examined iterative maps, providing specific references to the tonal characteristics that they can capture.

We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of 𝕂^k, where 𝕂 is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.

The Zwanzig's procedure (1960) for the description of the system of coupled harmonic oscillators is applied to the chain of interacting oscillations in order to find the adsorption power function, which is then determined by two terms: (i) the classical term proportional to the radio-frequency function squared and (ii) the additional term linear with respect to the radio-frequency magnetic field amplitude. From the physical point of view the first term is usually considered in oscillator effect. The second term found in the present paper via Zwanzig's procedure seems to be induced by fluctuations due to stochastic distributions of the oscillatory precession phases. It reflects well the chaos as described in a fractal approach of the first author (2012, paper joint with M. Nowak-Kȩpczyk and O. Suzuki).

Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.

Two X-band microwave Gunn oscillators have been separately operated with "below threshold" dc bias voltages under the influence of an injected weak RF field in their respective cavities to generate chaotic oscillations. The output of one such chaotic Gunn oscillator is injected into the other through a controllable coupling network to explore the possibility of synchronization between two oscillators. We establish through numerical simulation that (i) two oscillators with identical design parameters attain a state of complete synchronization and (ii) two oscillators with slightly different design parameters attain a state of generalized synchronization for reasonable value of coupling strengths. The occurrence of generalized synchronization has been proved through the "auxiliary" slave system approach of nonlinear analysis. Results of hardware experiments are incorporated to qualitatively support the observations made through numerical simulation.

In this paper, we present a new approach for finding periodic orbits in dynamical systems modeled by differential equations. It is based on the Homotopy Analysis Method (HAM) but it differs from the usual way it is applied. We apply HAM to construct approximations of a formal series solution of the equation. These approximations exist for any value of the frequency. They can be obtained by choosing a suitable linear operator and allowing the initial conditions to vary freely. Then we study the behavior of the obtained expressions as a function of the frequency. This procedure allows us to find those values of the frequency for which the series converges and therefore to find the periodic orbits. We show several examples of application of the proposed method. Mathematica and MATCONT were applied for all the calculations.

We establish the results on the existence of pullback attractors for nonautonomous functional differential equations with multiple delays appearing within nonlinear Lipschitz terms. The results are complementary to recently presented findings in [Caraballo & Kiss, 2013a, 2013b], and they extend the class of nonlinearities to which existence results can be established by improving on a condition presented in [Caraballo & Kiss, 2013b].

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

We consider a single Chua's circuit and a system of a unidirectionally coupled n-Chua's circuits driven by a biharmonic signal with two widely different frequencies ω and Ω, where Ω ≫ ω. We show experimental evidence for vibrational resonance in the single Chua's circuit and undamped signal propagation of a low-frequency signal in the system of n-coupled Chua's circuits where only the first circuit is driven by the biharmonic signal. In the single circuit, we illustrate the mechanism of vibrational resonance and the influence of the biharmonic signal parameters on the resonance. In the n(=75)-coupled Chua's circuits enhanced propagation of low-frequency signal is found to occur for a wide range of values of the amplitude of the high-frequency input signal and coupling parameter. The response amplitude of the ith circuit increases with i and attains a saturation. Moreover, the unidirectional coupling is found to act as a low-pass filter.

The relationship between period-doubling bifurcations and Feigenbaum's constants has been studied for nearly 40 years and this relationship has helped uncover many fundamental aspects of universal scaling across multiple nonlinear dynamical systems. This paper will combine information entropy with symbolic dynamics to demonstrate how period doubling can be defined using these tools alone. In addition, the technique allows us to uncover some unexpected, simple estimates for Feigenbaum's constants which relate them to log 2 and the golden ratio, φ, as well as to each other.