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 economic dynamics: Chaotic saddle, crisis and intermittency

Complex economic dynamics is studied by a forced oscillator model of business cycles. The technique of numerical modeling is applied to characterize the fundamental properties of complex economic systems which exhibit multiscale and multistability behaviors, as well as coexistence of order and chaos. In particular, we focus on the dynamics and structure of unstable periodic orbits and chaotic saddles within a periodic window of the bifurcation diagram, at the onset of a saddle-node bifurcation and of an attractor merging crisis, and in the chaotic regions associated with type-I intermittency and crisis-induced intermittency, in non-linear economic cycles. Inside a periodic window, chaotic saddles are responsible for the transient motion preceding convergence to a periodic or a chaotic attractor. The links between chaotic saddles, crisis and intermittency in complex economic dynamics are discussed. We show that a chaotic attractor is composed of chaotic saddles and unstable periodic orbits located in the gap regions of chaotic saddles. Non-linear modeling of economic chaotic saddle, crisis and intermittency can improve our understanding of the dynamics of financial intermittency observed in stock market and foreign exchange market. Characterization of the complex dynamics of economic systems is a powerful tool for pattern recognition and forecasting of business and financial cycles, as well as for optimization of management strategy and decision technology.     

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.     

Density of first Poincaré returns,periodic orbits,and Kolmogorov–Sinai entropy

It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.     

The energy resources system with parametric perturbations and its hyperchaos control

This paper analyses the dynamic behavior of an energy resources system with parametric perturbations. By adding the small sinusoidal perturbations to the three-dimensional energy resource system established for two regions of China, the autonomous system becomes the non-autonomous system which has richer dynamical behaviors. Periodic, chaotic and hyperchaos behaviors are discovered in the system by means of Lyapunov exponents and bifurcation diagrams. A new energy resources hyperchaos attractor is obtained, which has not yet been reported in present literature. Furthermore, effective non-autonomous feedback controllers are designed for stabilizing hyperchaos to unstable periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results. This study will be instructive for the energy resource demand-supply in some regions of China.     

A directed weighted complex network for characterizing chaotic dynamics from time series

We propose a reliable method for constructing a directed weighted complex network (DWCN) from a time series. Through investigating the DWCN for various time series, we find that time series with different dynamics exhibit distinct topological properties. We indicate this topological distinction results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Furthermore, we associate different aspects of dynamics with the topological indices of the DWCN, and illustrate how the DWCN can be exploited to detect unstable periodic orbits of different periods. Examples using time series from classical chaotic systems are provided to demonstrate the effectiveness of our approach.     

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.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Crisis of interspike intervals in Hodgkin–Huxley model

The bifurcations of the chaotic attractor in a Hodgkin–Huxley (H–H) model under stimulation of periodic signal is presented in this work, where the frequency of signal is taken as the controlling parameter. The chaotic behavior is realized over a wide range of frequency and is visualized by using interspike intervals (ISIs). Many kinds of abrupt undergoing changes of the ISIs are observed in different frequency regions, such as boundary crisis, interior crisis and merging crisis displaying alternately along with the changes of external signal frequency. And there are logistic-like bifurcation behaviors, e.g., periodic windows and fractal structures in ISIs dynamics. The saddle-node bifurcations resulting in collapses of chaos to period-6 orbit in dynamics of ISIs are identified.     

Pulsive feedback control of a quarter car model forced by a road profile

We examine the strange chaotic attractor and its unstable periodic orbits for a one degree of freedom nonlinear oscillator with a non-symmetric potential that models a quarter car forced by the road profile. We propose an efficient method of chaos control that stabilizes these orbits using a pulsive feedback technique. A discrete set of pulses is able to transfer the system from one periodic state to another.     

A new four-dimensional energy resources system and its linear feedback control

This paper reports a new four-dimensional energy resources chaotic system. The system is obtained by adding a new variable to a three-dimensional energy resource demand–supply system established for two regions of China. The dynamics behavior of the system will be analyzed by means of Lyapunov exponents and bifurcation diagrams. Linear feedback control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. Numerical simulations are presented to show these results.     

Homoclinic Tangles, Phase Locking, and Chaos in a Two-Frequency Perturbation of Duffing's Equation

Summary. We study a two-frequency perturbation of Duffing's equation. When the perturbation is small, this system has a normally hyperbolic invariant torus which may be subjected to phase locking. Applying a version of Melnikov's method for multifrequency systems, we detect the occurrence of transverse intersection between the stable and unstable manifolds of the invariant torus. We show that if the invariant torus is not subjected to phase locking, then such a transverse intersection yields chaotic dynamics. When the invariant torus is subjected to phase locking, the situation is different. In this case, there exist two periodic orbits which are created in a saddle-node bifurcation. Using another version of Melnikov's method for slowly varying oscillators, we also give conditions under which the stable and unstable manifolds of the periodic orbits intersect transversely and hence chaotic dynamics may occur. Our results reveal that when the invariant torus is subjected to phase locking, chaotic dynamics resulting from transverse intersection between its stable and unstable manifolds may be interrupted. Received November 18, 1993; final revision received September 9, 1997; accepted October 27,1997     

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.     

Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\R^{2}_+$ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.     

Global analysis of boundary and interior crises in an elastic impact oscillator

The crisis phenomena of a Duffing–Van der Pol oscillator with a one-side elastic constraint are studied by the composite cell coordinate system method in this paper. By computing the global properties such as attractors, basins of attraction and saddles, the vivid evolutionary process of two kinds of crises: boundary crisis and interior crisis are shown. The boundary crisis is resulted by the collision of a chaotic attractor and a periodic saddle on the basin boundary. It is observed that there are two types of interior crises. One is caused by the collision of a chaotic attractor and a chaotic saddle within the interior of basin of attraction. The other one occurs because a period attractor collides with a chaotic saddle within the interior of basin of attraction. The saddles of system play an important role in the crisis process. The results show that this method is an efficient tool to perform the global analysis of elastic impact oscillators.     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

Generation and suppression of a new hyperchaotic nonlinear model with complex variables

In this paper, we introduce a new hyperchaotic complex Chen model. This hyperchaotic complex system is constructed by adding a complex nonlinear term to the third equation of the chaotic complex Chen system with consideration it’s all variables are complex. The new system is a 6-dimensional continuous real autonomous hyperchaotic system. The properties of this system including invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and hyperchaotic behavior are studied. Different forms of hyperchaotic complex Chen systems are constructed. We suppress the hyperchaotic behavior of our system via passive control method by using one complex controller. The hyperchaotic attractors of the new system are converted to its unstable trivial fixed point and tracked to its unstable non trivial fixed points and periodic orbits. Block diagrams of our system are designed by using Matlab/Simulink after and before the suppression process to ensure the validity of the analytical results.     

Pitchfork bifurcation for non-autonomous interval maps

In this work, we investigate attracting periodic orbits for non-autonomous discrete dynamical systems with two maps using a new approach. We study some types of bifurcation in these systems. We show that the pitchfork bifurcation plays an important role in the creation of attracting orbits in families of alternating systems with negative Schwarzian derivative and it is central in the geometry of the bifurcation diagrams.