Chaos In Nonlinear Dynamical Systems Research Articles

Mid-infrared hyperchaos of interband cascade lasers

Chaos in nonlinear dynamical systems is featured with irregular appearance and with high sensitivity to initial conditions. Near-infrared light chaos based on semiconductor lasers has been extensively studied and has enabled various applications. Here, we report a fully-developed hyperchaos in the mid-infrared regime, which is produced from interband cascade lasers subject to the external optical feedback. Lyapunov spectrum analysis demonstrates that the chaos exhibits three positive Lyapunov exponents. Particularly, the chaotic signal covers a broad frequency range up to the GHz level, which is two to three orders of magnitude broader than existed mid-infrared chaos solutions. The interband cascade lasers produce either periodic oscillations or low-frequency fluctuations before bifurcating to hyperchaos. This hyperchaos source is valuable for developing long-reach secure optical communication links and remote chaotic Lidar systems, taking advantage of the high-transmission windows of the atmosphere in the mid-infrared regime.

Entanglement-Structured LSTM Boosts Chaotic Time Series Forecasting.

Traditional machine-learning methods are inefficient in capturing chaos in nonlinear dynamical systems, especially when the time difference between consecutive steps is so large that the extracted time series looks apparently random. Here, we introduce a new long-short-term-memory (LSTM)-based recurrent architecture by tensorizing the cell-state-to-state propagation therein, maintaining the long-term memory feature of LSTM, while simultaneously enhancing the learning of short-term nonlinear complexity. We stress that the global minima of training can be most efficiently reached by our tensor structure where all nonlinear terms, up to some polynomial order, are treated explicitly and weighted equally. The efficiency and generality of our architecture are systematically investigated and tested through theoretical analysis and experimental examinations. In our design, we have explicitly used two different many-body entanglement structures—matrix product states (MPS) and the multiscale entanglement renormalization ansatz (MERA)—as physics-inspired tensor decomposition techniques, from which we find that MERA generally performs better than MPS, hence conjecturing that the learnability of chaos is determined not only by the number of free parameters but also the tensor complexity—recognized as how entanglement entropy scales with varying matricization of the tensor.

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

Abstract In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is presented for a better understanding of global behaviors an...

Periodic motions and chaos in nonlinear dynamical systems

Periodic motions and chaos in nonlinear dynamical systems

On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in a periodically forced, time-delayed, twin-well Duffing oscillator are investigated. Such period-3 motions and corresponding bifurcation trees may or may not be embedded in chaos in the bifurcation trees of period-1 motions to chaos. The bifurcation trees for period-3 motions to chaos in such a time-delayed, twin-well Duffing oscillator are obtained analytically. From the finite discrete Fourier series, harmonic frequency-amplitude characteristics for period-3 to period-6 motions are analyzed. From the analytical prediction, numerical illustrations of period-3 and period-6 motions in the time-delayed, twin-well Duffing oscillator are completed. The complexity of period-3 motions to chaos in nonlinear dynamical systems is strongly dependent on the distributions and quantity levels of harmonic amplitudes. For a very slowly varying excitation, the infinite bifurcation trees of period-3 motions to chaos can be achieved once the excitation amplitude approaches to infinity.

Towards Infinite Bifurcation Trees of Period-1 Motions to Chaos in a Time-delayed, Twin-well Duffing Oscillator

In this paper, bifurcation trees of periodic motions to chaos in a periodically forced, time-delayed, twin-well Duffing oscillator are pre- dicted by a semi-analytical method. The twin-well Duffing oscilla tor is extensively used in physics and engineering. The bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is very significant for determine motion complexity. Thus, the bi furcation trees for periodic motions to chaos in such a time-delayed, twin-well Duffing oscillator are obtained analytically. From the fi nite discrete Fourier series, harmonic frequency-amplitude character istics for period-1 to period-4 motions are analyzed. The stability and bifurcation behaviors of the time-delayed Duffing oscillator are different from the non-time-delayed Duffing oscillator. From the an alytical prediction, numerical illustrations of periodic motions in the time-delayed, twin-well Duffing oscillator are completed. The com plexity of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. As a slowly varying excitation becomes very slow, the excitation amplitude will approach infinity for the infinite bifurcation trees of period-1 motion to chaos. Thus infinite bifurca tion trees of period-1 motion to chaos can be obtained.

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.

On Analytical Routes to Chaos in Nonlinear Systems

In this paper, the analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems are reviewed. Briefly discussed are the traditional analytical methods including the Lagrange stand form, perturbation methods, and method of averaging. A brief literature survey of approximate methods in application is completed, and the weakness of current existing approximate methods is also discussed. Based on the generalized harmonic balance, the analytical solutions of periodic motions in nonlinear dynamical systems with/without time-delay are reviewed, and the analytical solutions for period-m motion to quasi-periodic motion are discussed. The analytical bifurcation trees of period-1 motion to chaos are presented as an application.

Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay

In this paper, the analytical dynamics of periodic flows to chaos in nonlinear dynamical systems is presented from the ideas of Luo (Continuous dynamical systems, Higher Education Press/L&H Scientific, Beijing/Glen Carbon, 2012)). The analytical solutions of periodic flows and chaos in autonomous systems are discussed through the generalized harmonic balance method, and the analytical dynamics of periodically forced nonlinear dynamical systems is presented as well. The analytical solutions of periodic motions in free and periodically forced vibration systems are presented. The similar ideas are extended to time-delayed nonlinear systems. The analytical solutions of periodic flows to chaos for time-delayed, nonlinear systems with/without periodic excitations are presented, and time-delayed, nonlinear vibration systems will be also discussed. The analytical solutions of periodic flows and chaos are independent of small parameters, which are different from the traditional perturbation methods. The methodology presented herein will provide the accurate analytical solutions of periodic motions to chaos in dynamical systems with/without time-delay. This approach can handle nonlinear dynamical systems with either single time-delay or multiple time-delays.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

GLOBAL TANGENCY AND TRANSVERSALITY OF PERIODIC FLOWS AND CHAOS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

Transition to chaos in nonlinear dynamical systems described by ordinary differential equations

We investigate scenarios that create chaotic attractors in systems of ordinary differential equations (Vallis, Rikitaki, Rossler, etc.). We show that the creation of chaotic attractors is governed by the same mechanisms. The Feigenbaum bifurcation cascade is shown to be universal, while subharmonic and homoclinic cascades may be complete, incomplete, or not exist at all depending on system parameters. The existence of a saddle-focus equilibrium plays an important and possibly decisive role in the creation of chaotic attractors in dissipative nonlinear systems described by ordinary differential equations.

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2 n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.

THE ONSET OF CHAOS IN NONLINEAR DYNAMICAL SYSTEMS DETERMINED WITH A NEW FRACTAL TECHNIQUE

A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.

Control applications in physics: From control of chaos to quantum control

The last decade of last previous century has given rise of considerable advances in control and synchronization of chaos in nonlinear dynamical systems. Moreover a growing interaction between theoretical physics and control theory has become apparent. In this plenary lecture these developments are surveyed and closely examined. Several case examples most characteristic for these developments are used to provide appropriate illustrations. The feedback is the key concept to all these novel developments. An attempt is being made to define and outline a potentially new discipline of cybernetic physics.

ON THE MATHEMATICAL CLARIFICATION OF THE SNAP-BACK-REPELLER IN HIGH-DIMENSIONAL SYSTEMS AND CHAOS IN A DISCRETE NEURAL NETWORK MODEL

We investigate the differences among several definitions of the snap-back-repeller, which is always regarded as an inducement to produce chaos in nonlinear dynamical system. By analyzing the norms in different senses and the illustrative examples, we clarify why a snap-back-repeller in the neighborhood of the fixed point, where all eigenvalues of the corresponding variable Jacobian Matrix are absolutely larger than 1 in norm, might not imply chaos. Furthermore, we theoretically prove the existence of chaos in a discrete neural networks model in the sense of Marotto with some parameters of the systems entering some regions. And the following numerical simulations and corresponding calculation, as concrete examples, reinforce our theoretical proof.

Transition to Chaos in Nonlinear Dynamical Systems Through a Subharmonic Cascade of Bifurcations of Two-Dimensional Tori

Transition to Chaos in Nonlinear Dynamical Systems Through a Subharmonic Cascade of Bifurcations of Two-Dimensional Tori

Diagnosing chaos in non-linear dynamical systems by trajectory predictions and innovation tests of the Kalman filter

We report the use of the Kalman filter to detect the onset of chaos in a non-linear system for which the model is exactly known. The procedure is based on predicting the trajectories of a non-linear dynamical system and testing the innovation sequence for the predicted trajectory using the local overall method tests (LOMT). This is applied to the Rossler–Wegmann model of the Zhabotinskii reaction system. The Rossler–Wegmann model is a three dimensional set of non-linear ordinary differential equations, the solution to which gives the time-evolution of the concentrations of the major species in the Zhabotinskii reaction. With suitable parameters and starting values of concentrations, stable, oscillatory and chaotic solutions may be found. Five thousand points were generated from the model, for each of the three species in the model, by fourth order Runge–Kutta integration. For a complex oscillatory case the LOMT showed no chaos, and when parameters that lead to chaos were used the LOMT quickly revealed the mismatch between predicted and actual trajectories. It is concluded that the Kalman filter with LOMT is a quick and accurate method of diagnosing chaos, which could be used in a monitoring and on-line controlling system for a chemical process.

A hypothesis concerning ‘quantal’ Hilbert space criterion of chaos in nonlinear dynamical systems

Based on the Hilbert space approach to the theory of nonlinear dynamical systems developed by the author, a hypothesis is formulated concerning the ‘quantal’ criterion for classical ordinary differential systems to exhibit chaotic behaviour.

Chaos in nonlinear dynamical systems exemplified by an R & D model

Nonlinear dynamical systems theory plays an increasing role in the mathematical analysis of economic problems. The main purpose of this paper is to present some basic concepts and results occuring in this context to the OR/MS community. After a brief review of a few papers on chaotic management systems, we present a time-discrete model which shows on the one hand the effects of a firm's R & D-activity on their turnover, and on the other hand takes into consideration the important role of the management for the firm's behaviour. The influence of the management is modeled by a dynamic nonlinear decision rule, which is well-known in behavioural decision theory. The counteracting effects of a firm's evolution and the reactions of the management may lead to erratic behaviour.