Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

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.     

Chaos behavior in the discrete Fitzhugh nerve system

The discrete Fitzhugh nerve systems obtained by the Euler method is investigated and it is proved that there exist chaotic phenomena in the sense of Marotto’s definition of chaos. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynarnical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits and intermittent chaos. The computations of Lyapunov exponents confirm the chaos behaviors. Moreover we also find a strange attractor having the self-similar ohit structure as that of Henon attractor.     

Universal quantification for deterministic chaos in dynamical systems

A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth, i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model predicts the following: (a) The phase space trajectory (strange attractor) when resolved as a function of the computer accuracy has intrinsic logarithmic spiral curvature with the quasiperiodic Penrose tiling pattern for the internal structure. (b) The universal constant for deterministic chaos is identified as the steady-state fractional round-off error k for each computational step and is equal to 1/τ² ( = 0.382) where τ is the golden mean. k being less than half accounts for the fractal (broken) Euclidean geometry of the strange attractor. (c) The Feigenbaum's universal constantsa and d are functions of k and, further, the expression 2a² = πd quantifies the steady-state ordered emergence of the fractal geometry of the strange attractor. (d) The power spectra of chaotic dynamical systems follow the universal and unique inverse power law form of the statistical normal distribution. The model prediction of (d) is verified for the Lorenz attractor and for the computable chaotic orbits of Bernoulli shifts, pseudorandom number generators, and cat maps.     

Dynamics of a two-frequency parametrically driven duffing oscillator

Summary We investigate the transition from two-frequency quasiperiodicity to chaotic behavior in a model for a quasiperiodically driven magnetoelastic ribbon. The model system is a two-frequency parametrically driven Duffing oscillator. As a driving parameter is increased, the route to chaos takes place in four distinct stages. The first stage is a torus-doubling bifurcation. The second stage is a transition from the doubled torus to a strange nonchaotic attractor. The third stage is a transition from the strange nonchaotic attractor to a geometrically similar chaotic attractor. The final stage is a hard transition to a much larger chaotic attractor. This latter transition arises as the result of acrisis, the characterization of which is one of our primary concerns. Numerical evidence is given to indicate that the crisis arises from the collision of the chaotic attractor with the stable manifold of a saddle torus. Intermittent bursting behavior is present after the crisis with the mean time between bursts scaling as a power law in the distance from the critical control parameter; τ ∼ (A-A_c)^-α. The critical exponent is computed numerically, yielding the value α=1.03±0.01. Theoretical justification is given for the computed critical exponent. Finally, a Melnikov analysis is performed, yielding an expression for transverse crossings of the stable and unstable manifolds of the crisis-initiating saddle torus.     

Endogenous chaotic dynamics in transitional economies

This paper examines a model of labor market dynamics in an economy undergoing transition from command socialism to market capitalism. State sector layoffs are modeled as a function of forecasts made by state planners of private sector wages where the laidoff workers are to be re-employed. The state switches between using a high information cost perfect forecast and a free naive forecast in a system that resembles a cobweb supply-demand model. Under certain specifications and parameter values chaotic dynamics are shown to endogenously emerge along with several other varieties of complex dynamics including strange attractors, coexistence of infinitely many stable cycles, cascades of infinitely many period doubling bifurcations and fractal basin boundaries between coexisting non-chaotic attractors.     

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.     

Chaos in a bienzymatic cyclic model with two autocatalytic loops

A general bienzymatic cyclic system including two autocatalytic loops is studied and used as a basic design principle for modelling extracellular matrix turnover. Using classical enzyme kinetic rates, the model is described by a set of four ordinary differential equations and numerically studied by bifurcation diagrams and Poincaré sections. We observe limit-cycle oscillations and chaotic behaviors arising from period-doubling cascades or intermittency. Chaotic oscillations originate from distinct strange attractors that undergo boundary and internal crisis. For some parameter values, the system presents several bistable areas, where a limit cycle coexists with another one or with a strange attractor. The dynamics are qualitatively modified when the weight of the autocatalytic loops on the system varies, resulting in the change in the number of attractors.     

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.     

A periodically forced,piecewise linear system,Part II: The fragmentation mechanism of strange attractors and grazing

In the first part of this work, the local singularity of non-smooth dynamical systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical systems, which will help us better understand chaotic motions in non-smooth dynamical systems.     

Unwrapping trajectories of a quasi-periodic forced oscillator can give maps of the real line with SNA

We start from a simple model of quasi-periodic forced oscillator and show how it is possible to find corresponding maps of the two dimension torus. One of these constructions is frequently used to analyse the apparition of strange non-chaotic attractors (SNA in brief). SNA are complex attractors, their trajectories seem to be chaotic but without the sensitivity to initial conditions. Some trajectories can be unwrapped from the torus giving a simple iteration of the real line. It leads us to suppose that iterations from a simple continuous increasing map with quasi-periodic displacement can exhibit SNA.     

The abundance of strange attractor in the families of nearby singular diffeomorphism

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor. Project Supported by Fund of National Science of China     

Chaos in a four-dimensional system consisting of fundamental lag elements and the relation to the system eigenvalues

A simple system consisting of a second-order lag element (a damped linear pendulum) and two first-order lag elements with piecewise-linear static feedback that has been derived from a power system model is presented. It exhibits chaotic behavior for a wide range of parameter values. The analysis of the bifurcations and the chaotic behavior are presented with qualitative estimation of the parameter values for which the chaotic behavior is observed. Several characteristics like scalability of the attractor and globality of the attractor-basin are also discussed.     

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.     

On grazing and strange attractors fragmentation in non-smooth dynamical systems

This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-smooth dynamic systems. The sufficient and necessary conditions for grazing bifurcations in non-smooth dynamic systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-smooth dynamical systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.     

Singularity Structure and Chaotic Behavior of the Homopolar Disk Dynamo

We study in great detail a system of three first-order ordinary differential equations describing a homopolar disk dynamo (HDD). This system displays a large variety of behaviors, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The nonintegrability aspects of the HDD system are studied by investigating analytically the singularity structure of the system in the complex domain. Coexisting attractors (including period-doubling sequence) and coexisting strange attractors appear in some parametric regimes. The gluing of strange attractors and the ungluing of a strange attractor are also shown to occur. A period of bifurcation leading to chaos, not observed for other chaotic systems, is shown to characterize the chaotic behavior in some parametric ranges. The limiting case of the Lorenz system is also studied and is related to HDD.     

Adaptation of homotopy analysis method for the numeric–analytic solution of Chen system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy analysis method (HAM) is presented, which is the multistage homotopy analysis method (MSHAM). The freedom of choosing the auxiliary linear operator and the auxiliary parameter are still present in the MSHAM. The solutions of the non-chaotic and the chaotic Chen system which is a three-dimensional system of ordinary differential equations with quadratic nonlinearities were obtained by MSHAM. Numerical comparisons between the MSHAM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for solving the non-linear chaotic and non-chaotic Chen system.     

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.     

Control of discrete-time chaotic systems via feedback linearization

An approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method.