Identification of chaos in a regenerative cutting process by the 0-1 test

We examine the regenerative cutting process by using a single degree of freedom non-smooth model with a friction component and a time delay term. Instead of the standard Lyapunov exponent calculations, we propose a statistical 0-1 test for chaos. This approach reveals the nature of the cutting process signaling regular or chaotic dynamics. We are able to show that regular or chaotic motion occur in the investigated model depending on the delay time. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

Computer-assisted analysis of chaos in a three-species food chain model

In this paper, a three-species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer-assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2-shift map. At last, we consider the impact of fear on this three-species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.      

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 in a new system with fractional order

The dynamics of fractional-order systems have attracted a great deal of attentions in recent years. With fractional order, the dynamics of a system which includes comprehensive dynamical behaviors, such as fixed point, periodic motion, chaotic motion, and transient chaos is studied numerically in this paper. It is known that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order found for this system to yield chaos is 2.43. The results are validated by the existence of a positive Lyapunov exponent. Period doubling routes to chaos in the fractional-order system are also obtained. Moreover, generation of a four-scroll chaotic attractor by the system is observed.     

Chaos control of chaotic dynamical systems using backstepping design

This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lü systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.     

Bifurcations of periodic solutions and chaos in Duffing-van der Pol equation with one external forcing

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.     

Horseshoes in chromosome’s attractors

In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincaré maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.     

动力系统实测数据的非线性混沌模型重构

动力系统实测非线性混沌数据的模型重构技术是相空间重构的重要内容。在判定了实测数据的非线性混沌特征,计算了实测数据的分维数,Lyapunov指数,并对其进行了本征值分解和噪声去除及确定其模型阶数以后,提出了一个动力系统实测数据的非线性混沌模型,给出了相应的模型参数辨识方法,并用其确立的混沌模型进行了预测工作,计算结果表明:模型参数辨识方法能迅速地将参数估计值带到多峰目标函数的全局最少值附近,然后再采用优化理论能较准确地求出模型的参数,用得到的混沌模型对系统进行预测工作其预测效果良好,且混沌时序不可能作长期预测。     

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.     

Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity

In this paper, we consider the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The amplitude equations are derived in the general case using the harmonic balance method. The model without discontinuity is first considered. The effects of the amplitude of the parametric modulation and some particular coefficients are found in the response curves. The transition to chaotic behavior is found using numerical simulations of the equations of motion. We find that chaos appears in the model between the quasi-periodic and periodic orbits when the amplitude of the external excitation E₀ vary. An adaptive Lyapunov control strategy enables us to drive the system from the chaotic states to a targeting periodic orbit. The effects of elasticity and damping on the dynamics of the self-sustained electromechanical system are also derived.     

Identification of chaos in a cutting process by the 0–1 test

We have examined the cutting process by using a two degrees of freedom non-smooth model with a friction component. Instead of the standard Lyapunov exponent treatment a statistical ‘0–1’ test based on the asymptotic properties of a non-harmonic Brownian motion chain has been successively applied to reveal the nature of the cutting process. In this test we calculated the control parameter K which is approaching asymptotically to 0 or 1 for regular and chaotic motions, respectively. The presented approach is independent on the integration procedure as we defined a characteristic distance between the points forming the time series used in the test separately.     

A parametric method of constructing Poincaré mappings in hydrodynamic systems

The plane-parallel motion of the particles of an incompressible medium reduces to an investigation of a Hamilton system. The stream function is a Hamilton function. A Hamilton function, which depends periodically on time and corresponds to the agitation of an incompressible medium in a domain which varies periodically with time, is considered. This agitation of the medium is due to dynamic chaos. The transition to dynamic chaos is described by investigating the location of the Lagrangian particles over time intervals which are multiples of the period (Poincaré points (PP)). The set of PP can be obtained using a Poincaré mapping in the phase flow. The method which has been developed is used to investigate the plane-parallel motion of the particles in an incompressible fluid in a thin layer, bounded by a flat bottom, rectilinear side walls and an upper boundary which is deformed according to a specified periodic law. The motion of the particles is determined from Hamilton's system of equations. The Hamiltonian (the stream function) is found in the thin-layer approximation and depends on two dimensionless parameters: the amplitude of deformation and the tangential velocity in the deforming boundary. The characteristic boundary, which separates the domain of the chaotic motion of the PP from the domain of ordered motion, is determined numerically in the domain of the two parameters. The topological structure of the phase trajectories up to the transition to chaotic conditions is analysed using the Poincaré mapping, found with an accuracy up to the third order with respect to the amplitude. The phase trajectories of the PP, found analytically, turn out to be close to the trajectories of the initial Hamilton system, determined numerically. The mapping found in the domain of the two dimensionless parameters enables one to determine, qualitatively, the boundary of the transition to chaos.     

Chaos, complex transients and noise: Illustration with a Kaldor model

In the present paper two-dimensional discrete Kaldor-type models are investigated. First, a sufficient condition for the existence of topological chaos of the model is derived analytically for a special parameter set. Second, the influences of noise on the Kaldor model are examined numerically. We show that noise may not only obscure the underlying structures, but also reveal the hidden structures, for example, the chaotic attractors near a window of chaos or the periodic attractors near a small chaotic parameter region.     

Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach

Based on stability theory of impulsive differential equation and new comparison theory of impulsive differential system, we study the chaos impulsive synchronization of two coupled chaotic systems using the unidirectional linear error feedback scheme. Some generic conditions of chaos impulsive synchronization of two coupled chaotic systems are derived, and to apply the conditions to typical chaotic system––the original Chua’s circuit. The example illustrates the effectiveness of the proposed result.     

Global chaos in a periodically forced, linear system with a dead-zone restoring force

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are introduced through switching planes pertaining to two constraints. The global periodic motions based on the Poincare mapping are determined, and the eigenvalue analysis for the stability and bifurcation of periodic motion is carried out. Global chaos in such a system is investigated numerically from the unstable global periodic motions analytically determined. The bifurcation scenario with varying parameters is presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed in this investigation.     

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.     

Dynamic analysis, controlling chaos and chaotification of a SMIB power system

The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called “swing equation”. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincaré maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued.