Complexity Of Chaos Research Articles

Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities

The local stability and existence of periodic motions in a periodically forced oscillator with multiple discontinuities are investigated. The complexity of periodic motions and chaos in such a discontinuous system is often caused by the passability, sliding, and grazing of flows to discontinuous boundaries. Therefore, the corresponding analytical conditions for such singular phenomena to discontinuous boundary are presented from the local singularity theory of discontinuous systems. To develop the mapping structures of periodic motions, basic mappings are introduced, and the sliding motion on the discontinuous boundary is described by a sliding mapping. A generalized mapping structure is presented for all possible periodic motions, and the local stability and bifurcations of periodic motions are discussed. From mapping structures, the switching points of periodic motions on the boundaries are predicted analytically. Two periodic motions are presented for illustrations of the passability, sliding, and grazing of periodic motions on the boundary.

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.

Complexity of chaos and quantum computation

This paper reviews recent work related to the interplay between quantum information and computation on the one hand and classical and quantum chaos on the other.First, we present several models of quantum chaos that can be simulated efficiently on a quantum computer. Then a discussion of information extraction shows that such models can give rise to complete algorithms including measurements that can achieve an increase in speed compared with classical computation. It is also shown that models of classical chaos can be simulated efficiently on a quantum computer, and again information can be extracted efficiently from the final wave function. The total gain can be exponential or polynomial, depending on the model chosen and the observable measured. The simulation of such systems is also economical in the number of qubits, allowing implementation on present-day quantum computers, some of these algorithms having been already experimentally implemented.The second topic considered concerns the analysis of errors on quantum computers. It is shown that quantum chaos algorithms can be used to explore the effect of errors on quantum algorithms, such as random unitary errors or dissipative errors. Furthermore, the tools of quantum chaos allows a direct analysis of the effects of static errors on quantum computers. Finally, we consider the different resources used by quantum information, and show that quantum chaos has some precise consequences on entanglement generation, which becomes close to maximal. For another resource, interference, a proposal is presented for quantifying it, enabling a discussion on entanglement and interference generation in quantum algorithms.

The Lempel-Ziv Complexity of Non-Stationary Chaos in Infinite Ergodic Cases

The large deviation properties of the Lempel-Ziv complexity are studied using a onedimensional non-hyperbolic chaos map called the “modified Bernoulli map”, where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/f ν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L 1 -function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/f spectral process is generated.

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.

Quantifying the complexity of chaos in multibasin multidimensional dynamics of molecular systems

AbstractThe simulated classical dynamics of a small molecule exhibiting self‐organizing behavior via a fast transition between two states is analyzed by calculation of the statistical complexity of the system. It is shown that the complexity of molecular descriptors such as atom coordinates and dihedral angles have different values before and after the transition. This provides a new tool to identify metastable states during molecular self‐organization. The highly concerted collective motion of the molecule is revealed. Low‐dimensional subspaces dynamics is found sensitive to the processes in the whole, high‐dimensional phase space of the system. © 2004 Wiley Periodicals, Inc. Complexity 10: 40‐46, 2004

PREY PREFERENCE BY A TOP PREDATOR AND THE STABILITY OF LINKED FOOD CHAINS

Recent theoretical studies have shown the potential for chaotic dynamics in simple three-species food chains. Most of these studies have focused on linear food chains, although natural food chains are seldom isolated from the surrounding food web. There is a growing awareness that food web dynamics can be strongly influenced by the behavior and movement of predators, energy, and nutrients across ecosystem and subecosystem boundaries. Motivated by observations from lakes, where the pelagic food web is often linked to the littoral food web by mobile predators, we constructed a simple model to evaluate the dynamics of two food chains linked by a top predator with prey preference. Linking the two food chains had no qualitative effect on model dynamics, although it did increase the density of the top predator. Instead, the prey preference of the top predator changed the system dynamics. We found a range of prey preferences that could eliminate chaos, dampen oscillations, and even produce point stability in a previously oscillatory system. The strength of prey preference required to produce a point attractor in a previously chaotic system was positively related to the dimension of chaos (a measure of the complexity of chaos). Our results suggest that, although chaos is possible in food webs, common processes like prey preference reduce the potential for chaos.

Communicating with chaos.

The use of chaos to transmit information is described. Chaotic dynamical systems, such as electrical oscillators with very simple structures, naturally produce complex wave forms. We show that the sym- bolic dynamics of a chaotic oscillator can be made to follow a desired symbol sequence by using small perturbations, thus allowing us to encode a message in the wave form. We illustrate this using a simple numerical electrical oscillator model. PACS numbers: 05.45.+b Much of the fundamental understanding of chaotic dy- namics involves concepts from information theory, a field developed primarily in the context of practical communi- cation. Concepts from information theory used in chaos include metric entropy, topological entropy, Markov par- titions, and symbolic dynamics (1). On the other hand, because of their exponential sensitivity, chaotic systems are often said to evolve randomly. This terminology is partially justified if one regards the information obtained by detailed observation of the chaotic orbit as being less significant than the statistical properties of the orbits. The object of this Letter is to show that we can use the close connection between the theory of chaotic systems and information theory in a way that is more than purely formal. In particular, we show that the recent realization that chaos can be controlled with small perturbations (2) can be utilized to cause the symbolic dynamics of a chaotic system to track a prescribed symbol sequence, thus allowing us to encode any desired message in the sig- nal from a chaotic oscillator. The natural complexity of chaos thus provides a vehicle for information transmission in the usual sense. Furthermore, we argue that this method of communication will often have technological advantages. Specifically, assume that there is an electrical oscillator producing a large amplitude chaotic signal that one wishes to use for communication. The so-called double