Autonomous strange nonchaotic oscillations in a system of mechanical rotators

We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

Analysis and circuitry realization of a novel three-dimensional chaotic system

In this paper a new three-dimensional chaotic system is introduced. Some basic dynamical properties are analyzed to show chaotic behavior of the presented system. These properties are covered by dissipation of system, instability of equilibria, strange attractor, Lyapunov exponents, fractal dimension and sensitivity to initial conditions. Through altering one of the system parameters, various dynamical behaviors are observed which included chaos, periodic and convergence to an equilibrium point. Eventually, an analog circuit is designed and implemented experimentally to realize the chaotic system.     

On the exponentially self-regulating population model

A new type of discrete dynamical systems model for populations, called an exponentially self-regulating (ESR) map, is introduced and analyzed in considerable detail for the case of two competing species. The ESR model exhibits many dynamical features consistent with the observed interactions of populations and subsumes some of the most successful discrete biological models that have been studied in the literature. For example, the well-known Tribolium model is an ESR map. It is shown that in addition to logistic dynamics – ranging from the very simple to manifestly chaotic one-dimensional regimes – the ESR model exhibits, for some parameter values, its own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is proved that ESR systems have twisted horseshoe with bending tail dynamics associated to an essentially global strange attractor for certain parameter ranges. The existence of a global strange attractor makes the ESR map more plausible as a model for actual populations than several other extant models, including the Lotka–Volterra map.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

Micro-chaos in digital control

Summary In this paper we analyze a model for the effect of digital control on one-dimensional, linearly unstable dynamical systems. Our goal is to explain the existence of small, irregular oscillations that are frequently observed near the desired equilibrium. We derive a one-dimensional map that captures exactly the dynamics of the continuous system. Using thismicro-chaos map, we prove the existence of a hyperbolic strange attractor for a large set of parameter values. We also construct an “instability chart” on the parameter plane to describe how the size and structure of the chaotic attractor changes as the parameters are varied. The applications of our results include the stick-and-slip motion of machine tools and other mechanical problems with locally negative dissipation.     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

On the nonlinear Poisson bracket arising in nonholonomic mechanics

Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all.     

The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

Dynamics of Perturbed Wavetrain Solutions to the Ginzburg-Landau Equation

The bifurcation structure and asymptotic dynamics of even, spatially periodic solutions to the time-dependent Ginzburg-Landau equation are investigated analytically and numerically. All solutions spring from unstable periodic modulations of a uniform wavetrain. Asymptotic states include limit cycles, two-tori, and chaotic attractors. Lyapunov exponents for some chaotic motions are obtained. These show the solution strange attractors to have a fractal dimension slightly greater than 3.     

Chaotic dynamics of a discrete prey–predator model with Holling type II

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.     

Duffing–van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit

We have constructed a simple second-order dissipative nonautonomous circuit exhibiting ordered and chaotic behaviour. This circuit is the well known Murali–Lakshmanan–Chua(MLC) circuit but with diode based nonlinear element. For chosen circuit parameters this circuit admits familiar MLC type attractor and also Duffing–van der Pol circuit type chaotic attractors. It is interesting to note that depending upon the circuit parameters the circuit shows both period doubling route to chaos and quasiperiodic route to chaos. In our study we have constructed two-parameter bifurcation diagrams in the forcing amplitude–frequency plane, one parameter bifurcation diagrams, Lyapunov exponents, 0–1 test and phase portrait. The performance of the circuit is investigated by means of laboratory experiments, numerical integration of appropriate mathematical model and explicit analytic studies.     

Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.     

Richness of chaotic dynamics in nonholonomic models of a celtic stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.     

Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts

For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.     

Simple polynomial classes of chaotic jerky dynamics

Third-order explicit autonomous differential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic long-time behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to these days, are known to show chaotic behavior in some parameter range. After deriving several analytical properties of these systems, we systematically determine the dependence of the long-time dynamical behavior on the system parameters by numerical evaluation of Lyapunov spectra. Some features of the systems that are related to the dependence on initial conditions are also addressed. The observed dynamical complexity of the two systems is discussed in connection with the existence of homoclinic orbits.     

Structure and properties of the attractor of a marine dynamical system

We investigate the properties of a marine dynamical system by means of time series of the sea-level height at four locations in the Saronicos Gulf in the Aegean Sea, Greece. In order to characterize the dynamics, we estimate the dimension of the underlying system attractor, and we compute its Lyapunov exponents. Dimension estimates indicate that the dynamics can be explained by a low-dimensional deterministic dynamical system. Lyapunov exponent estimates further substantiate the above conclusion, while at the same time, indicate that the dynamical system is a rather nonuniform chaotic one.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.