Terms Of Unstable Periodic Orbits Research Articles

Network analysis of chaotic systems through unstable periodic orbits.

A chaotic motion can be considered an irregular transition process near unstable periodic orbits embedded densely in a chaotic set. Therefore, unstable periodic orbits have been used to characterize properties of chaos. Statistical quantities of chaos such as natural measures and fractal dimensions can be determined in terms of unstable periodic orbits. Unstable periodic orbits that can provide good approximations to averaged quantities of chaos or turbulence are also known to exist. However, it is not clear what type of unstable periodic orbits can capture them. In this paper, a model for an irregular transition process of a chaotic motion among unstable periodic orbits as nodes is constructed by using a network. We show that unstable periodic orbits which have lots of links in the network tend to capture time averaged properties of chaos. A scale-free property of the degree distribution is also observed.

Unstable periodic orbits in plane Couette flow with the Smagorinsky model

We aim at a description of the logarithmic velocity profile of wall turbulence in terms of unstable periodic orbits (UPOs) for plane Couette flow with a Smagorinsky-type eddy viscosity model. We study the bifurcation structure with respect to the Smagorinsky constant, arising from the gentle UPO reported by Kawahara and Kida [1] for the Navier-Stokes (NS) equation. We find that the obtained UPOs in the large eddy simulation (LES) system connect to those in the NS system, and that the gentle UPO in the LES system is an edge state branch whose stable manifold separates LES turbulence from an LES ‘laminar’ state. As the Reynolds number decreases this solution arises as the saddle solution of the saddle-node bifurcation. Meanwhile, the mean and root-mean-square velocity profiles of the node solution of the LES gentle UPO are in good agreement with those of LES turbulence.

Breakdown mechanisms of normally hyperbolic invariant manifolds in terms of unstable periodic orbits and homoclinic/heteroclinic orbits in Hamiltonian systems

We analyze the mechanism of breakdown of normally hyperbolic invariant manifolds (NHIMs) based on unstable periodic orbits, homoclinic and heteroclinic orbits in NHIMs for Hamiltonian systems. First, we classify the breakdown mechanism in terms of the characteristic multipliers (the eigenvalues of the phase space Jacobian matrix) of unstable periodic orbits in the NHIM and elucidate the classification for systems of three degrees of freedom. Second, we also present an index that provides a sufficient condition under which the NHIM ceases to exist in more global regions along the homoclinic and heteroclinic orbits in the NHIM. We demonstrate local and global features of the breakdown of the NHIM for a hydrogen atom in crossed electric and magnetic fields. Our analysis of unstable periodic orbits of shorter periods indicates that local features of the breakdown are highly inhomogeneous, depending on each periodic orbit. This manifests an inherent dynamical feature for systems of more than two degrees of freedom. Contrastingly, our analysis of homoclinic orbits indicates that the energy at which the NHIM breaks down does not depend on each homoclinic orbit. This result suggests the existence of a global breakdown of the NHIM behind these homoclinic orbits since these homoclinic orbits run through broader phase space regions.

Manifold structures of unstable periodic orbits and the appearance of periodic windows in chaotic systems.

Manifold structures of the Lorenz system, the Hénon map, and the Kuramoto-Sivashinsky system are investigated in terms of unstable periodic orbits embedded in the attractors. Especially, changes of manifold structures are focused on when some parameters are varied. The angle between a stable manifold and an unstable manifold (manifold angle) at every sample point along an unstable periodic orbit is measured using the covariant Lyapunov vectors. It is found that the angle characterizes the parameter at which the periodic window corresponding to the unstable periodic orbit finishes, that is, a saddle-node bifurcation point. In particular, when the minimum value of the manifold angle along an unstable periodic orbit at a parameter is small (large), the corresponding periodic window exists near (away from) the parameter. It is concluded that the window sequence in a parameter space can be predicted from the manifold angles of unstable periodic orbits at some parameter. The fact is important because the local information in a parameter space characterizes the global information in it. This approach helps us find periodic windows including very small ones.

Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model

The theory of chaotic dynamical systems gives many tools that can be used in climate studies. The widely used ones are the Lyapunov exponents, the Kolmogorov entropy and the attractor dimension characterizing global quantities of a system. Another potentially useful tool from dynamical system theory arises from the fact that the local analysis of a system probability distribution function (PDF) can be accomplished by using a procedure that involves an expansion in terms of unstable periodic orbits (UPOs). The system measure (or its statistical characteristics) is approximated as a weighted sum over the orbits. The weights are inversely proportional to the orbit instability characteristics so that the least unstable orbits make larger contributions to the PDF. Consequently, one can expect some relationship between the least unstable orbits and the local maxima of the system PDF. As a result, the most probable system trajectories (or 'circulation regimes' in some sense) may be explained in terms of orbits. For the special classes of chaotic dynamical systems, there is a strict theory guaranteeing the accuracy of this approach. However, a typical atmospheric model may not qualify for these theorems. In our study, we will try to apply the idea of UPO expansion to the simple atmospheric system based on the barotropic vorticity equation of the sphere. We will check how well orbits approximate the system attractor, its statistical characteristics and PDF. The connection of the most probable states of the system with the least unstable periodic orbits will also be analysed.

Periodic-orbit determination of dynamical correlations in stochastic processes

It is shown that the large-deviation statistical quantities of the discrete-time, finite-state Markov process P_{n+1};{(j)}= summation _{k=1};{N}H_{jk}P_{n};{(k)} , where P_{n};{(j)} is the probability for the j state at the time step n and H_{jk} is the transition probability, completely coincide with those from the Kalman map corresponding to the above Markov process. Furthermore, it is demonstrated that, by using simple examples, time correlation functions in finite-state Markov processes can be well described in terms of unstable periodic orbits embedded in the equivalent Kalman maps.

Coexistence of synchronized and desynchronized patterns in coupled chaotic dynamical systems

We analyse the synchronous chaos and its stability in coupled map lattices and coupled nonlinear oscillators. The existence of desynchronized patterns such as standing wave like spatiotemporal periodic patterns within the critical size limit is identified for some sets of random initial conditions. The emergence of such patterns is explained using the concept of controlling of chaos. In the case of coupled map lattices, an expression for standing wave like pattern is given in terms of unstable periodic orbits of the isolated map. Finally, the role of initial conditions on the existence of multiple stable states in both coupled map lattices and coupled oscillators is studied.

Determination of Chaotic Dynamical Correlations in Terms of Unstable Periodic Orbits

For low-dimensional chaotic systems, we find that time correlation functions can be accurately approximated by a single unstable periodic orbit. The method of this determination consists of the following two steps. First, the time correlation functions are expanded in terms of static correlation functions by making use of the expansion method recently proposed by one of the authors [H. Fujisaka, Prog. Theor. Phys. 114 (2005), 1]. Second, the static quantities are approximated in terms of an unstable periodic orbit embedded in the strange attractor. Thus the dynamical correlation functions in chaotic systems can be determined in terms of an unstable periodic orbit. Furthermore, applying this method to low-dimensional chaotic models, we prove the usefulness of the present approach.

Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits

Abstract. A description of a deterministic chaotic system in terms of unstable periodic orbits (UPO) is used to develop a method of an a priori estimate of the sensitivity of statistical averages of the solution to small external influences. This method allows us to determine the forcing perturbation which maximizes the norm of the perturbation of a statistical moment of the solution on the attractor. The method was applied to the barotropic ocean model in order to determine the perturbation of the wind field which provides the greatest perturbation of the model's climate. The estimates of perturbations of the model's time mean solution and its mean variance were compared with directly calculated values. The comparison shows that some 20 UPOs is sufficient to realize this approach and to obtain a good accuracy.

Sensitivity of attractor to external influences: approach by unstable periodic orbits

A description of a deterministic chaotic system in terms of unstable periodic orbits (UPOs) is used to develop a method of an a priori estimate of the sensitivity of statistical averages of the solution to small external influences. This method allows us to determine the forcing perturbation which maximizes the norm of the perturbation of a statistical moment of the solution on the attractor. The method was applied to the Lorenz model. The estimates of perturbations of two statistical moments were compared with directly calculated values. The comparison shows that some 100 UPOs are sufficient to realize this approach and get a good accuracy. The linear approach remains valid up to rather high norms of the forcing perturbation.

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

Using nuclear spins, radio waves, sodium atoms, and laser light to investigate spatiotemporal nonlinear systems

Within the wide field of nonlinear dynamics we investigate temporal and spatial behavior of electromagnetic systems. A strange type of laser, the nuclear magnetic resonance (NMR) laser, shows truly chaotic behavior and is therefore ideally suited to analyze experimentally and theoretically a variety of temporal nonlinear effects. Of particular interest is the analysis of its strange attractors in terms of unstable periodic orbits. For the extension of our research from the NMR laser (representing a purely temporal system described by the Bloch-Kirchhoff ordinary differential equations) to spatial and spatiotemporal systems, we chose, as a three-dimensional dynamic system, polarized laser beams interacting nonlinearly with sodium atoms. Among other effects, we have observed beam bouncing, beam splitting, and beam switching. This can be well described by partial differential equations for beam propagation derived from the Schrödinger equation and the Maxwell equations. An intuitive explanation is given in terms of intensity and polarization patterns formed by optical-pumping-induced mutual refractive index modifications between polarized resonant laser beams.

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) $H_{q}$ relevant to the fluctuation spectrum and poles of the order-$q$ power spectrum is proposed. The ``partition function'' is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.

Unstable periodic orbits and transport properties of nonequilibrium steady states.

Nonequilibrium steady-state flows governed by time-reversible equations of motion are described in terms of unstable periodic orbits. Important properties, such as transport coefficients and the multifractal spectrum, are related to the difference in probability of observing periodic orbits and their time reverses. We apply the theory to the periodic Lorentz gas driven by a constant external field and numerically calculate the conductivity.

Periodic orbits as the skeleton of classical and quantum chaos

A description of a low-dimensional deterministic chaotic system in terms of unstable periodic orbits (cycles) is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos, comparable to the familiar perturbation expansions for nearly integrable systems. The infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a Selberg product form, dominated by the short cycles, and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits. Methods for implementing this computation for finite subshift dynamics are introduced.

Unstable periodic orbits and prediction.

A hierarchical approximation of a generic chaotic attractor can be formulated in terms of unstable periodic orbits. We demonstrate the possibility of extracting the most dominant unstable periodic orbits from a measurement of a time-continuous flow. Since unstable periodic orbits not only represent the static properties of the system but also dominate the dynamics, they can be used to fit models of the flow and to predict the orbit. We show that predictions for the Roessler system using unstable periodic orbits extracted from a time series of moderate length are significantly better than those from other approaches.

The spectrum of the period-doubling operator in terms of cycles

The eigenvalue spectrum of the period-doubling operator is evaluated both in terms of unstable periodic orbits and by a diagonalization at the operator. Cycles up to length ten yield the twelve leading eigenvalues; direct diagonalization yields some 50 eigenvalues. The Feigenbaum 6, the Hausdorff dimension and the escape rate for the period-doubling repeller are evaluated to high accuracy. In this letter we apply the technique for the extraction of correlation exponents of (I) to evaluation of the spectrum of the period-doubling renormalization operator. The basic idea of relating the Feigenbaum constant 6 to the scaling (or the presentation) function is due to Sullivan (2), and is implemented as an eigenvalue spectrum calcula- tion in (3-51. We refer the reader to (3, 41 for an introduction to the cycle expansions in general, and their application to the evaluation of the stability, the dimension and the escape rate of the period-doubling repeller in particular. The Fredholm determinant evaluation method and the transfer operator diagonalization ( 11 used here are superior to the method of locating zeros of finite products of dynamical 6 functions (6) of (3, 4, 7, 81, and in that sense the present letter supersedes the above references. The repeller we study here is the non-wandering set of the period-doubling presenta- tion function (9)

Regular and irregular potential scattering

The authors present evidence for irregular behaviour in potential scattering and discuss a possible explanation in terms of unstable periodic orbits.