Nonlinear vibration and characteristics of flexible plate-strips with non-symmetric boundary conditions

Regular and chaotic vibrations together with bifurcations of flexible plate-strips with non-symmetric boundary conditions, are investigated through the Bubnov–Galerkin method and a finite difference method of error O(h⁴). Particular attention is paid to non-symmetric boundary conditions. Lyapunov exponents are estimated via Bennetin’s method. Some new examples of routes from regular to chaotic dynamics, and within chaotic dynamics are illustrated and discussed. The phase transitions from chaos to hyperchaos, and a novel phenomenon of a shift from hyperchaos to hyperhyper chaos is also reported.     

基于Lyapunov指数的高炉铁水[Si]预报

通过对国内两座中型高炉冶炼过程的[S i]时间序列的混沌分析,计算出相应的Lyapunov指数谱.由最大Lyapunov指数为正,定量的说明了两座高炉冶炼过程具有混沌性,并估计了两座高炉冶炼过程[S i]可预测的时间尺度.同时根据最大Lyapunov指数,建立了高炉冶炼过程[S i]预报模型,取得了较好的结果.     

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.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

The brief time-reversibility of the local Lyapunov exponents for a small chaotic Hamiltonian system

We consider the local (instantaneous) Lyapunov spectrum for a four-dimensional Hamiltonian system. Its stable periodic motion can be reversed for long times. Its unstable chaotic motion, with two symmetric pairs of exponents, cannot. In the latter case reversal occurs for more than a thousand fourth-order Runge–Kutta time steps, followed by a transition to a new set of paired Lyapunov exponents, unrelated to those seen in the forward time direction. The relation of the observed chaotic dynamics to the Second Law of Thermodynamics is discussed.     

Bifurcation analysis of bacteria and bacteriophage coexistence in the presence of bacterial debris

The dynamics of bacteria and bacteriophage coexistence in the presence of bacterial debris, in a marine environment, was studied using a system of delay differential equations (DDE). The system exhibits a rich variety of behavior in terms of two control parameters values: the bacteriophage burst size β, and the lysing time delay τ. Limit cycles of various periodicity, quasiperiodicity, period doubling, chaotic bands and toroidal chaos were identified using basic tools of non-linear dynamics analysis: first return maps, Poincaré sections, Fourier spectrum, and largest Lyapunov exponents.     

Solution and dynamics analysis of a fractional‐order hyperchaotic system

Numerical solution and chaotic behaviors of the fractional‐order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional‐order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional‐order chaotic system is designed. Dynamics of the fractional‐order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C₀ complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order q is small. It lays a foundation for the practical application of the fractional‐order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Symmetric Jacobian for local Lyapunov exponents and Lyapunov stability analysis revisited

The stability analysis introduced by Lyapunov and extended by Oseledec provides an excellent tool to describe the character of nonlinear n-dimensional flows by n global exponents if these flows are stationary in time. However, here we discuss two shortcomings: (a) the local exponents fail to indicate the origin of instability where trajectories start to diverge. Instead, their time evolution contains a much stronger chaos than the trajectories, which is only eliminated by integrating over a long time. Therefore, shorter time intervals cannot be characterized correctly, which would be essential to analyse changes of chaotic character as in transients. (b) Although Oseledec uses an n dimensional sphere around a point x to be transformed into an n dimensional ellipse in first order, this local ellipse has not yet been evaluated. The aim of this contribution is to eliminate these two shortcomings. Problem (a) disappears if the Oseledec method is replaced by a frame with a ‘constraint’ as performed by Rateitschak and Klages (RK) [Rateitschak K, Klages R, Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering. Phys Rev E 2002;65:036209/1–11]. The reasons why this method is better will be illustrated by comparing different systems. In order to analyze shorter time intervals, integrals between consecutive Poincaré points will be evaluated. The local problem (b) will be solved analytically by introducing the ‘symmetric Jacobian for local Lyapunov exponents’ and its orthogonal submatrix, which enable to search in the full phase space for extreme local separation exponents. These are close to the RK exponents but need no time integration of the RK frame. Finally, four sets of local exponents are compared: Oseledec frame, RK frame, symmetric Jacobian for local Lyapunov exponents and its orthogonal submatrix.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a ϕ6 potential

This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ⁶ potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ⁶ potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

Complex dynamics and control of arms race

The aim of this paper is to show that asymmetric, nonlinear armament strategies may lead to chaotic motion in a discrete-time Richardson-type model on the arms race between two rival nations. Local bifurcation analysis reveals that ‘complicated’ dynamics will only occur if neither nation has an absolute advantage over the other one with respect to its level of armament and its capability to keep up the expenditures on armament. The calculation of Lyapunov exponents supports the existence of chaos. Since transitions to chaos can be identified with transitions to war, we use the Ott-Grebogi-Yorke-algorithm to stabilize the arms race model in the chaotic regime and improve the system's performance by making very small time-dependent changes of a parameter under control.     

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.     

Hyperchaotic attractors from a linearly controlled Lorenz system

In this paper, a four-dimensional (4D) continuous-time autonomous hyperchaotic system with only one equilibrium is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the second equation of the 3D Lorenz system. Some complex dynamical behaviors of the hyperchaotic system are investigated, revealing many interesting properties: (i) existence of periodic orbit with two zero Lyapunov exponents; (ii) existence of chaotic orbit with two zero Lyapunov exponents; (iii) chaos depending on initial value $w_0$; (iv) chaos with only one equilibrium; and (v) hyperchaos with only one equilibrium. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are derived and studied.     

Chaotic behavior of an autocatalytic reaction with mutation

A conceptual model of an autocatalytic reaction in a continuous stirred tank reactor in which the autocatalyst undergoes mutation has been examined. The model consists of a simple set of three ordinary differential equations with seven design parameters. It is shown that self-sustained chaotic behavior can occur in this system. The regions of chaos are entered and exited according to period-doubling cascades or through a process involving intermittency. Lyapunov exponents are calculated to confirm the chaotic behavior.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

Suppression and generation of chaos for a three-dimensional autonomous system using parametric perturbations

In this paper, we attempt to suppress or generate chaos in the newly presented Lü system using parametric perturbation. We find that this method not only suppresses chaotic behavior, but also induces chaos in non-chaotic parameter ranges. When we add the small sinusoidal perturbations, the system becomes four-dimension. From the calculation of Lyapunov exponents, we discover hyperchaos in the perturbed system, which has not yet been reported before.