Number Of Unstable Periodic Orbits Research Articles

Chaos control applied to piezoelectric vibration-based energy harvesting systems

Chaotic behavior presents intrinsic richness due to the existence of an infinity number of unstable periodic orbits (UPOs). The possibility of stabilizing these periodic patterns with a small amount of energy makes this kind of response interesting to various dynamical systems. Energy harvesting has as a goal the use of available mechanical energy by promoting a conversion into electrical energy. The combination of these two approaches may establish autonomous systems where available environmental mechanical energy can be employed for control purposes. Two different goals can be defined as priority, allowing a change between them: vibration reduction and energy harvesting enhancement. This work deals with the use of harvested energy to perform chaos control. Both control actuation and energy harvesting are induced employing piezoelectric materials, in a simultaneous way. A bistable piezomagnetoelastic structure subjected to harmonic excitations is investigated as a case study. Numerical simulations show situations where it is possible to perform chaos control using only the energy generated by the harvesting system.

Chaotification as a Means of Broadband Energy Harvesting With Piezoelectric Materials

Component miniaturization and reduced power requirements in sensors have enabled growth in the field of low-power ambient vibration energy harvesting. This work aims to increase bandwidth and power output beyond current techniques by inducing chaotic nonlinear phenomena and applying a low-power controller based on the method of Ott, Grebogi, and Yorke (OGY) to stabilize a chosen periodic orbit. Previously, researchers used a nonlinear piezomagnetoelastic beam in search of a large amplitude broadband voltage response, but chaos was strictly avoided. These large amplitude responses can deteriorate over time into low energy chaotic oscillations. Including chaos as a desirable property allows small perturbations to alter the behavior of a system dramatically, improving the dynamic response for energy harvesting. The nonlinear piezomagnetoelastic beam element described by a Duffing oscillator is extended to embrace chaotic motion more actively. By driving motion along a chaotic attractor, even single frequency excitation results in a theoretically infinite number of unstable periodic orbits that can be stabilized using small control inputs. The chosen orbit will be accessible from a large range of excitation frequencies and can be dynamically changed in real-time, potentially expanding the bandwidth of operation.

Universalities in the chaotic generalized Moore & Spiegel equations

By investigating the topology of chaotic solutions to the generalized Moore and Spiegel equations, we address the question how the solutions of nonlinear dynamical systems are dependent on the nature of the nonlinearity. In these generalized jerk equations, the single nonlinear term has a parity which depends on unu̇. The system has an inversion symmetry when n is even and no symmetry property when n is odd. It is shown that the topology of chaotic solutions only depends on the parity of n, that is, on the symmetry properties and not on the degree of nonlinearity. The value of n only affects the possibility to develop the chaotic solution, that is, to increase the number of unstable periodic orbits within the attractor.

Stabilizing the unstable periodic orbits of a hybrid chaotic system using optimal control

In this paper, we are interested in the control of a chaotic hybrid system with an application to Chua’s system. It is known that chaotic attractors contain an infinite number of unstable periodic orbits (UPO) with different lengths, our idea consists in stabilizing a predetermined orbit of a given length by using an optimal control method. Our approach is to determine the switching instants from one subsystem to the other while minimizing the difference between two successive orbits. Should the switchings be state dependent, as is the case for the well known Chua’s circuit, then our approach consists in perturbing the switching boundaries such that the system trajectory hits those boundaries at the specified instants. Numerical simulations illustrating the efficiency of the proposed method are presented.

Studying a Chaotic Spiking Neural Model

Dynamics of a chaotic spiking neuron model are being studied mathematically and experimentally. The Nonlinear Dynamic State neuron (NDS) is analysed to further understand the model and improve it. Chaos has many interesting properties such as sensitivity to initial conditions, space filling, control and synchronization. As suggested by biologists, these properties may be exploited and play vital role in carrying out computational tasks in human brain. The NDS model has some limitations; in thus paper the model is investigated to overcome some of these limitations in order to enhance the model. Therefore, the models parameters are tuned and the resulted dynamics are studied. Also, the discretization method of the model is considered. Moreover, a mathematical analysis is carried out to reveal the underlying dynamics of the model after tuning of its parameters. The results of the aforementioned methods revealed some facts regarding the NDS attractor and suggest the stabilization of a large number of unstable periodic orbits (UPOs) which might correspond to memories in phase space.

Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems

Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking ``clustering'' phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the ``mixed'' role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

Harnessing quantum transport by transient chaos

Chaos has long been recognized to be generally advantageous from the perspective of control. In particular, the infinite number of unstable periodic orbits embedded in a chaotic set and the intrinsically sensitive dependence on initial conditions imply that a chaotic system can be controlled to a desirable state by using small perturbations. Investigation of chaos control, however, was largely limited to nonlinear dynamical systems in the classical realm. In this paper, we show that chaos may be used to modulate or harness quantum mechanical systems. To be concrete, we focus on quantum transport through nanostructures, a problem of considerable interest in nanoscience, where a key feature is conductance fluctuations. We articulate and demonstrate that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns. Experimentally, this can be achieved by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible potential barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. To understand the physical mechanism of our scheme, we develop a theory based on analyzing the spectrum of the generalized non-Hermitian Hamiltonian that includes the effect of leads, or electronic waveguides, as self-energy terms. As the escape rate of the underlying non-attracting chaotic set is increased, the imaginary part of the complex eigenenergy becomes increasingly large so that pointer states are more difficult to form, making smoother the conductance-fluctuation pattern.

Time-delayed feedback control of chaos in a GaAs/AlGaAs heterostructure

A theoretical model has been developed to study nonlinear behaviors in a GaAs/AlGaAs heterostructure. We show that the system can exhibit chaotic oscillations under transverse magnetic fields and an electric field. The time-delayed feedback method was applied to stabilize the unstable periodic orbits (UPOs) embedded in the chaotic attractor. A bifurcation specified by a successive decrease in the number of UPOs with delayed time and feedback strength was revealed, indicating the stabilization of UPOs under feedback control. Noticeably, the introduction of a feedback perturbation will sometimes induce chaotic states that do not exist in the unperturbed system.

Global view on a nonlinear oscillator subject to time-delayed feedback control

We apply time-delayed feedback control to stabilise unstable periodic orbits of an amplitude-phase oscillator. The control acts on both, the amplitude and the frequency of the oscillator, and we show how the phase of the control signal influences the dynamics of the oscillator. A comprehensive bifurcation analysis in terms of the control phase and the control strength reveals large stability regions of the target periodic orbit, as well as an increasing number of unstable periodic orbits caused by the time delay of the feedback loop. Our results provide insight into the global features of time-delayed control schemes.

Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

Periodic Orbits and Chaotic Sets in a Low-Dimensional Model for Shear Flows

We consider the dynamics of a low-dimensional model for turbulent shear flows. The model is based on Fourier modes and describes sinusoidal shear flow, in which fluid between two free-slip walls experiences a sinusoidal body force. The model contains nine modes, most of which have a direct hydrodynamical interpretation. We analyze the stationary states and periodic orbits for the model for two different domain sizes. Several kinds of bifurcations are identified, including saddle-node bifurcations, a period doubling cascade, and Hopf bifurcations of the periodic orbits. For both domain sizes, long-lived transient chaos appears to be associated with the presence of a large number of unstable periodic orbits. For the smaller minimal flow unit domain, it is found that a periodic solution is stable over a range of Reynolds numbers, and its bifurcations lead to the existence of a chaotic attractor. The model illustrates many phenomena observed and speculated to exist in the transition to turbulence in linearly stable shear flows.

ATTENTION-LOCKED COMPUTATION WITH CHAOTIC NEURAL NETS

We review a neural network model based on chaotic dynamics [Babloyantz & Lourenço, 1994, 1996] and provide a detailed discussion of its biological and computational relevance. Chaos can be viewed as a "reservoir" containing an infinite number of unstable periodic orbits. In our approach, the periodic orbits are used as coding devices. By considering a large enough number of them, one can in principle expand the information processing capacity of small or moderate-size networks. The system is most of the time in an undetermined state characterized by a chaotic attractor. Depending on the type of an external stimulus, the dynamics is stabilized into one of the available periodic orbits, and the system is then ready to process information. This corresponds to the system being driven into an "attentive" state. We show that, apart from static pattern processing, the model is capable of dealing with moving stimuli. We especially consider in this paper the case of transient visual stimuli, which has a clear biological relevance. The advantages of chaos over more regular regimes are discussed.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators.

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.

Chaos–hyperchaos transition in coupled Rössler systems

Many of the nonlinear high-dimensional systems have hyperchaotic attractors. Typical trajectory on such attractors is characterized by at least two positive Lyapunov exponents. We provide numerical evidence that chaos–hyperchaos transition in six-dimensional dynamical system given by flow can be characterized by the set of infinite number of unstable periodic orbits embedded in the attractor as it was previously shown for the case of two coupled discrete maps.

Chaos-hyperchaos transition

The chaos-hyperchaos transition occurs when the second Lyapunov exponent becomes positive. We argue that this transition is mediated by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. Bifurcations of unstable periodic orbits occur in the neighborhood of the chaos-hyperchaos transition point where we observe unstable variable dimensionality. We give evidence that the chaos-hyperchaos transition is initiated by (i) the saddle-repeller bifurcation of a particular unstable periodic orbit usually of low period, (ii) the appearance of a repelling node in the saddle-node bifurcation, after which the chaotic attractor becomes riddled, or (iii) the absorption of the repeller (unstable node or focus) originally located out of the attractor by the growing attractor.

A time-dependent quantum mechanical investigation of dynamical resonances in three-dimensional HeH2+ and HeHD+ systems

Bound and quasibound states of HeH2+ and HeHD+ in three dimensions, for zero total angular momentum, have been computed using a time-dependent quantum mechanical approach. Time evolution of a carefully chosen wave packet in the interaction region is followed and the time correlation function evaluated and its Fourier transform obtained. The resulting eigenvalue spectrum and the corresponding eigenfunctions are examined to characterize the nature of the dynamical resonances for the system. It becomes clear that at low energies the quasibound states can be assigned readily in terms of local modes. While some of the higher energy state eigenfunctions resemble the hyperspherical modes, a large number of them cannot be assigned easily, suggesting irregular dynamics, in keeping with a large number of unstable periodic orbits known for the system.

Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits

We investigate how the transition to chaos with multiple positive Lyapunov exponents can be characterized by the set of infinite number of unstable periodic orbits embedded in the chaotic invariant set. We argue and provide numerical confirmation that the transition is generally accompanied by a nonhyperbolic behavior: unstable dimension variability. As a consequence, the Lyapunov exponents, except for the largest one, pass through zero continuously.

Estimating generating partitions of chaotic systems by unstable periodic orbits

An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.

Periodic-orbit theory of the blowout bifurcation

This paper presents a theory for characterization of the blowout bifurcation by periodic orbits. Blowout bifurcation in chaotic systems occurs when a chaotic attractor, lying in some symmetric invariant subspace, becomes transversely unstable. We present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. Our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits, in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits.

Characterization of blowout bifurcation by unstable periodic orbits

Blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor, lying in some invariant subspace, becomes transversely unstable. We establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor. We argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits . There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. @S1063-651X~97!50902-0# PACS number~s!: 05.45.1b Recently, a novel type of bifurcation has been discovered in chaotic dynamical systems @1,2#. This is the so-called ‘‘blowout bifurcation’’ that occurs in dynamical systems with a symmetric invariant subspace. Let S be the invariant subspace in which there is a chaotic attractor. Since S is invariant, initial conditions in S result in trajectories that remain in S forever. Whether the chaotic attractor in S is also an attractor in the full phase space depends on the sign of the largest Lyapunov exponent L’ computed for trajectories in S with respect to perturbations in the subspace T which is transverse to S. When L’ is negative, S attracts trajectories transversely in the vicinity of S and, hence, the chaotic attractor in S is an attractor in the full phase space. If L’ is positive, trajectories in the neighborhood of S are repelled away from it and, consequently, the attractor in S is transversely unstable and it is hence not an attractor in the full phase space. Blowout bifurcation occurs when L’ changes from negative to positive values. There are distinct physical phenomena associated with the blowout bifurcation. For example, near the bifurcation point where L ’ is negative, if there are other attractors in the phase space, then typically, the basin of the chaotic attractor in S is riddled @3#. When L ’ is slightly positive, if there are no other attractors in the phase space, the dynamics in the transverse subspace T exhibits an extreme type of temporally intermittent bursting behavior, the on-off intermittency @4,5#. Recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems @6#. In the study of chaos theory, it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics @7‐9#. Periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems. For example, the period-doubling bifurcation @10# and the saddlenode bifurcation are bifurcations of periodic orbits. Catastrophic events in chaotic systems such as crises @11# and basin boundary metamorphoses @12# are triggered by collision of periodic orbits, usually of low period, embedded in different dynamical invariant sets. The birth of Wada basin boundaries, meaning common boundaries of more than two basins of attraction, is caused by a saddle-node bifurcation on the basin boundary @13#. More recent study indicates that the riddling bifurcation, bifurcation that gives birth to a riddled basin, is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in S @14#. In view of the role of periodic orbits played in these major bifurcations, it is desirable to study the blowout bifurcation by periodic orbits. In this regard, Ashwin, Buescu, and Stewart have noticed that as a system parameter changes towards the blowout bifurcation point, more and more atypical invariant measures become transversely unstable @2#. At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable. In this paper, we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace S .I n particular, we argue that near the bifurcation, there exist two groups of periodic orbits S s and S u , each having an infinite number of members, one transversely stable and another transversely unstable, respectively. The sign of the largest transverse Lyapunov exponent L’ is determined by the relative weights of S s and S u : L’ is negative ~positive! when S s ~S u! weighs over S u ~S s!. ~A precise definition of the ‘‘weights’’ will be described in the sequel. ! At the bifurcation, the weights of S s and S u are balanced. In contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits @10‐14#, blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits . The num