Periodic Saddle Research Articles

Control of chaos by occasional bang-bang.

The stabilization of a periodic saddle orbit (the target orbit) in a strange attractor is usually achieved by the application of a sequence of parameter perturbations designed to place the system state in the stable manifold of the target, whereupon it evolves unperturbed to the target orbit. Controls of this type originated with the method of Ott, Grebogi, and Yorke (OGY), and usually require a continuously variable parameter for map based control. Bang-bang control is a method whereby control is achieved by the application of a fixed, or several different fixed perturbations, rather than a continuous range of perturbations, and generally requires a flexible scheduling. If we have available only fixed perturbation levels, and can apply control only at regular intervals and for fixed durations, standard OGY will not work. We demonstrate a method that will control maps and continuous systems at a surface of section, albeit imprecisely, with a single fixed perturbation of fixed duration. We call this occasional bang bang.

Characterization of the parameter-mismatching effect on the loss of chaos synchronization

We investigate the effect of the parameter mismatch on the loss of chaos synchronization in coupled one-dimensional maps. Loss of strong synchronization begins with a first transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor (SCA), and then the SCA becomes weakly stable. Because of local transverse repulsion of the periodic repellers embedded in the weakly stable SCA, a typical trajectory may have segments of arbitrary length that have positive local transverse Lyapunov exponents. Consequently, the weakly stable SCA becomes sensitive with respect to the variation of the mismatching parameter. To quantitatively characterize such parameter sensitivity, we introduce a quantifier, called the parameter sensitivity exponent (PSE). As the local transverse repulsion of the periodic repellers strengthens, the value of the PSE increases. In terms of these PSEs, we also characterize the parameter-mismatching effect on the intermittent bursting and basin riddling occurring in the regime of weak synchronization.

Effect of asymmetry on the loss of chaos synchronization.

We investigate the effect of asymmetry of coupling on the bifurcation mechanism for the loss of synchronous chaos in coupled systems. It is found that only when the symmetry-breaking pitchfork bifurcations take part in the process of the synchronization loss for the case of symmetric coupling, the asymmetry changes the bifurcation scenarios of the desynchronization. For the case of weak coupling, pitchfork bifurcations of asynchronous periodic saddles are replaced by saddle-node bifurcations, while for the case of strong coupling, pitchfork bifurcations of synchronous periodic saddles transform to transcritical bifurcations. The effects of the saddle-node and transcritical bifurcations for the weak asymmetry are similar to those of the pitchfork bifurcations for the symmetric-coupling case. However, with increasing the "degree" of the asymmetry, their effects change qualitatively, and eventually become similar to those for the extreme case of unidirectional asymmetric coupling.

New Riddling Bifurcation in Asymmetric Dynamical Systems

We investigate the bifurcation mechanism for the loss of transverse stability of the chaotic attractor in an invariant subspace in an asymmetric dynamical system. It is found that a direct transition to global riddling occurs through a transcritical contact bifurcation between a periodic saddle embedded in the chaotic attractor on the invariant subspace and a repeller on its basin boundary. This new bifurcation mechanism differs from that in symmetric dynamical systems. After such a riddling bifurcation, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and typical power-law scaling is found.

Mechanism for the riddling transition in coupled chaotic systems

We investigate the loss of chaos synchronization in coupled chaotic systems without symmetry from the point of view of bifurcations of unstable periodic orbits embedded in the synchronous chaotic attractor (SCA). A mechanism for direct transition to global riddling through a transcritical contact bifurcation between a periodic saddle embedded in the SCA and a repeller on the boundary of its basin of attraction is thus found. Note that this bifurcation mechanism is different from that in coupled chaotic systems with symmetry. After such a riddling transition, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and thus typical power-law scaling is found.

TRACKING SUSTAINED CHAOS

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO 2 laser.

On the transition from local regular to global irregular fluctuations

We present a general framework for understanding the transition from local regular to global irregular (chaotic) behaviour of nonlinear dynamical models in discrete time. The fundamental mechanism is the unfolding of quadratic tangencies between the stable and the unstable manifolds of periodic saddle points. To illustrate the relevance of the presented methods for analysing globally a class of dynamic economic models, we apply them to the infinite horizon model of Woodford (1988), (J. Economic Theory, 40, 128–137), amended by Grandmont et al. (1998), (J. Economic Theory, 80) to account for capital-labour substitution, so as to explain the appearance of irregular fluctuations, through homoclinic bifurcations, when parameter values are ‘far’ away from local bifurcation points .

FURTHER STUDY OF INTERIOR CRISES IN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Crises in systems of ordinary differential equations are investigated by means o f Generalized Cell Mapping Digraph (GCMD) method. We show that a boundary crisis results from a collision between a chaotic attractor and a periodic saddle on i ts basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the origin al chaotic attractor in phase space. We focus here on a sudden change in the siz e of a chaotic attractor, namely an interior crisis. We demonstrate that at an i nterior crisis the chaotic attractor collides with a chaotic saddle within its b asin of attraction. This chaotic saddle is an invariant and nonattracting set an d resembles the new portion of the larger chaotic attractor just after the inter ior crisis. We also investigate the origin and evolution of the chaotic saddle. The local refining procedures of persistent and transient self-cycling sets are given.

Finding all periodic orbits of maps using Newton methods: sizes of basins

For a diffeomorphism F on R 2, it is possible to find periodic orbits of F of period k by applying Newton’s method to the function F k − I, where I is the identity function. (We actually use variants of Newton’s method which are more robust than the traditional Newton’s method.) For an initial point x, we iterate Newton’s method many times. If the process converges to a point p which is a periodic point of F, we say x is in the Newton basin of p for period k, denoted by B(p,k) . We investigate the size of the Newton basin and how it depends on p and k. In order to understand the basins of high period orbits, we choose p a periodic point of F with period k, then we investigate basins B(p,nk) for n=1,2,3,… . We show that if p is an attracting orbit, then there is an open neighborhood of p that is in all the Newton basins B(p,nk) for all n. If p is a repelling periodic point of F, it is possible that p is the only point which is in all of the Newton basins B(p,nk) for all n. It is when p is a periodic saddle point of F that the Newton basin has its most interesting behavior. Our numerical data indicate that the area of the basin of a periodic saddle point p is proportional to λ c where λ is the magnitude of the unstable eigenvalue of DF k ( p) and c is approximately −1 ( c≈−0.84 in Fig. 5). For long periods ( k more than about 20), many orbits of F have λ so large that the basins are numerically undetectable. Our main result states that if p is a saddle point of F, the intersection of Newton basins B(p,nk) of p includes a segment of the local stable manifold of p.

GEOMETRY AND ORDER OF CHAOS

In the beginning a brief introduction to a symmetry increasing as well as to quantified geometrical structures of chaotic attractors have been given. Then, the research is focused on the Chossat–Golubitsky map. A saddle-node bifurcation on the boundary of the umbrella shape set of saddles, crises of two peculiar structures of saddles and an extremely high order of infinitely many flip and regular saddle orbits have been reported. The geometrical properties of hyperchaos and its relation to the Kakeya–Besicovitch and the Apollonian packing have been illustrated and discussed. A fractal organization of the successive periodic saddles and fractal normal forms have been described.

The frustrated and compositional nature of chaos in small Hopfield networks

Frustration in a network described by a set of ordinary differential equations induces chaos when the global structure is such that local connectivity patterns responsible for stable oscillatory behaviours are intertwined, leading to mutually competing attractors and unpredictable itinerancy among brief appearance of these attractors. Frustration destabilizes the network and provokes an erratic `wavering' among the periodic saddle orbits which characterize the same network when it is connected in a non-frustrated way. The characterization of chaos as some form of unpredictable `wavering' among repelling oscillators is rather classical but the originality here lies in the identification of these oscillators as the stable regimes of the `non-frustrated' network. In this paper, a simple and small 6-neuron Hopfield network is treated, observed and analyzed in its chaotic regime. Given a certain choice of the network parameters, chaos occurs when connecting the network in a specific way (said to be `frustrated') and gives place to oscillatory regimes by suppressing whatever connection between two neurons. The compositional nature of the chaotic attractor as a succession of brief appearances of orbits (or parts of orbits) associated with the non-frustrated networks is evidenced by relying on symbolic dynamics, through the computation of Lyapunov exponents, and by computing the autocorrelation coefficients and the spectrum.

Optimal targeting of chaos

Standard graph theoretic algorithms are applied to chaotic dynamical systems to identify orbits that are optimal relative to a prespecified cost function. We reduce the targeting problem to the problem of finding optimal paths through a graph. Numerical experiments on one-dimensional maps suggest that periodic saddle orbits of low period are typically less expensive to target (relative to a family of smooth cost functions) than periodic saddle orbits of high period.

Maintaining Chaos in High Dimensions

In dynamical systems, as a parameter is varied past a critical value, a chaotic attractor may be destroyed by a crisis. This attractor is replaced by a chaotic transient, which eventually leads to a different attractor. We present a method for maintaining chaotic dynamics after the crisis. The model, formulated for arbitrary dimensions, directs the phase space trajectory toward a target region near the periodic saddle orbit that mediates the crisis. It is used to maintain chaos numerically in the Ikeda map and experimentally in a magnetoelastic ribbon.

Infinitely Many Periodic Attractors for Holomorphic Maps of 2 Variables

An important development in the study of discrete dynamical systems was Newhouse's use of persistent homoclinic tangencies to show that a set of C2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors, or sinks [6], where large refers to a residual subset of an open set of diffeomorphisms. In the present paper, we obtain this result for various spaces of holomorphic maps of two variables. Newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency [7]. More recently, Palis and Viana extended this latter result to higher dimensions when the stable manifold has codimension one [10], and Romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks [12]. In each case, however, the construction reduces to the study of intersecting Cantor sets in the line: under an appropriate projection, the stable and unstable manifolds of a basic set are mapped to Cantor sets in the line, and a tangency between these manifolds corresponds to a point of intersection of the Cantor sets. The generic unfolding of tangencies then gives rise to periodic attractors or saddles. This reduction to linear Cantor sets depends heavily on the fact that there is only one expanding eigenvalue. Even Romero's result for higher codimension stable manifolds involves a reduction to this case. In the holomorphic setting, eigenvalues come in conjugate pairs (from a real point of view), so this reduction is not possible. Instead, stable and unstable manifolds are Riemann surfaces, and after extending the stable and unstable manifolds of a basic set to foliations, these foliations will be tangent in a (real) 2-dimensional disk, and the stable and unstable manifolds correspond to Cantor sets in this disk. Hence, persistent tangencies between basic sets correspond to the stable intersection of two Cantor sets in the plane.

Determination of global stability of steady-state solutions of a beam system with discontinuous support using manifolds ∗

This paper deals with the global stability of the long term dynamics of nonlinear mechanical systems under periodic excitation. Generally, the boundaries of the basins of attraction are formed by the stable manifolds of unstable periodic solutions. These stable manifolds are the set of initial conditions of trajectories which approach an unstable periodic saddle solution. Because these are the only trajectories which do not approach an attractor, in general the stable manifolds are the boundaries of the basins of attraction. In this paper manifolds are calculated of a beam system supported by a one-sided spring in order to identify the global stability of the coexisting attractors. The numerical results are compared with experimental results.

Characterization and detection of parameter variations of nonlinear mechanical systems

The present study applies the recently developed ideas in experimental system modeling to both characterize the behavior of simple mechanical systems and detect variations in their parameters. First, an experimental chaotic time series was simulated from the solution of the differential equation of motion of a mechanical system with clearance. From the scalar time series, a strange attractor was reconstructed optimally by the method of delays. Optimal reconstructions of the attractors can be achieved by simultaneously determining the minimal necessary embedding dimension and the proper delay time. Periodic saddle orbits were extracted from the chaotic orbit and their eigenvalues were calculated. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents for the steady state motion. An analysis of the associated one dimensional delay map, obtained from the chaotic time series, is made to determine the allowable periodic orbits and to yield an estimate of the topological entropy for the positive Lyapunov exponent. Sensitivity of the positions of the low order unstable periodic orbits (orbits of short period) of a chaotic attractor is used as a basis for detection of parameter variations in another unsymmetric bilinear system. For the experimental scalar time series generated by the dynamical system as a parameter varies, the chaotic attractors were again optimally reconstructed using the method of delays. The parameter variations were detected by the changes in location of the unstable periodic orbits extracted from the reconstructed attractors of the experimental scalar time series.

Controlling Chaos of Forced Pendulum by OGY Method.

It has already been verified that the OGY method is useful for confining chaotic motion such as the Henon map or Lorentz attractor to a periodic orbit that is arbitrarily chosen. In this paper, chaos of a forced pendulum and its control by the OGY method are discussed. The forced pendulum has more complicated chaotic motion than the above-mentioned chaos. First, periodic orbits of the forced pendulum are explored in order to find desired periodic orbits and saddles on a Poincare section, the OGY method is applied, and then some aspects of the OGY method on the forced pendulum are presented.

Metamorphoses: Sudden jumps in basin boundaries

In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are calledbasin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.

Infinitely many co-existing sinks from degenerate homoclinic tangencies

The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families { F t } \{ {F_t}\} of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, P t ; F t n ( P t ) = P t {P_t};\;F_t^n({P_t}) = {P_t} , and | det D F t n ( P t ) | > 1 |\det DF_t^n({P_t})| > 1 . We also require the stable and unstable manifolds of P t {P_t} to form homoclinic tangencies as the parameter t t varies through t 0 {t_0} . Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values t t near t 0 {t_0} . We show that there are parameter values t t near t 0 {t_0} at which F t {F_t} has infinitely many co-existing periodic sinks.

Characterization of an experimental strange attractor by periodic orbits

We describe a general procedure to locate periodic saddle orbits in a chaotic attractor reconstructed from experimental data. The method is applied to data from a Belousov-Zhabotinskii chemical reaction. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents. An analysis of the next amplitude map determines the allowable periodic orbits and yields an estimate of the topological entropy.