Unstable dimension variability in coupled chaotic systems

Route to hyperchaos in a system of coupled oscillators with multistability

This work presents the results of a detailed experimental study into the transition between synchronized, low-dimensional, and unsynchronized, high-dimensional dynamics using a system of coupled electronic chaotic oscillators. Novel data analysis techniques have been employed to reveal that a hyperchaotic attractor can arise from the amalgamation of two nonattracting sets. These originate from initially multistable low-dimensional attractors which experience a smooth transition from low- to high-dimensional chaotic behavior, losing stability through a bubbling bifurcation. Numerical techniques were also employed to verify and expand on the experimental results, giving evidence on the locally unstable invariant sets contained within the globally stable hyperchaotic attractor. This particular route to hyperchaos also results in the possibility of phenomena (such as unstable dimension variability) that can be a major obstruction to shadowing and predictability in chaotic systems.

Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

Chaotic dynamical systems with two or more attractors lying on invariant subspaces may, provided certain mathematical conditions are fulfilled, exhibit intermingled basins of attraction: Each basin is riddled with holes belonging to basins of the other attractors. In order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest (two-species competition with extinction) we have characterized quantitatively the intermingled basins using periodic-orbit theory and scaling laws. The latter results agree with a theoretical prediction from a stochastic model, and also with an exact result for the scaling exponent we derived for the specific class of models investigated. We discuss the consequences of the scaling laws in terms of the predictability of a final state (extinction of either species) in an ecological experiment.

Experimental study of the transitions between synchronous chaos and a periodic rotating wave

In this work we characterize experimentally the transition between periodic rotating waves and synchronized chaos in a ring of unidirectionally coupled Lorenz oscillators by means of electronic circuits. The study is complemented by numerical and theoretical analysis, and the intermediate states and their transitions are identified. The route linking periodic behavior with synchronous chaos involves quasiperiodic behavior and a type of high-dimensional chaos known as chaotic rotating wave. The high-dimensional chaotic behavior is characterized, and is shown to be composed actually by three different behaviors. The experimental study confirms the robustness of this route.

Method for measuring unstable dimension variability from time series

Many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity. One of the possible violations of this condition is the presence of unstable dimension variability (UDV), i.e., the existence in a chaotic attractor of sets of unstable periodic orbits, each with a different number of expanding directions. It has been shown that the presence of UDV poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems (the shadowing time). In this work we propose a method to detect the presence of UDV in real systems from time series measurements. Variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor. We show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region.

Multi-scale continuum mechanics: from global bifurcations to noise induced high-dimensional chaos

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.

Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability

Severe obstruction to shadowing of computer-generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability. That is, when the dimension of the unstable eigenspace changes along a trajectory in the invariant set, no true trajectory of reasonable length can be found to exist near any numerically generated trajectory. An important quantity characterizing the shadowability of numerical trajectories is the shadowing time, which measures for how long a trajectory remains valid. This time depends sensitively on initial condition. Here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on system details but the small-time algebraic behavior appears to be universal. We describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors.

Validity of numerical trajectories in the synchronization transition of complex systems

We investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly nonhyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization state. There are potentially severe consequences of these facts on the validity of the computer-generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties.

Universal and nonuniversal features in shadowing dynamics of nonhyperbolic chaotic systems with unstable-dimension variability

An important quantity characterizing the shadowability of computer-generated trajectories in nonhyperbolic chaotic system is the shadowing time, which measures for how long a numerical trajectory remains valid. This time depends sensitively on an initial condition. Here, we show that for nonhyperbolic systems with unstable-dimension variability, the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on the system details but the small-time algebraic behavior appears to be universal.

Shadowability of statistical averages in chaotic systems

We ask whether statistical averages in chaotic systems can be computed or measured reliably under the influence of noise. Situations are identified where the invariance of such averages breaks down as the noise amplitude increases through a critical level. An algebraic scaling law is obtained which relates the change of the averages to the noise variation. This breakdown of shadowability of statistical averages, as characterized by the algebraic scaling law, can be expected in both low- and high-dimensional chaotic systems.

Chaotic bursting at the onset of unstable dimension variability

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

Shadowing breakdown and large errors in dynamical simulations of physical systems

Simulations play a crucial role in the modern study of physical systems. A major open question for long dynamical simulations of physical processes is the role of discretization and truncation errors in the outcome. A general mechanism is described that can cause extremely small noise inputs to result in errors in simulation statistics that are several orders of magnitude larger. A scaling law for the size of such errors in terms of the noise level and properties of the dynamics is given. This result brings into question trajectory averages that are computed for systems with particular dynamical behaviors, in particular, systems that exhibit fluctuating Lyapunov exponents or unstable dimension variability.

Mechanisms for the development of unstable dimension variability and the breakdown of shadowing in coupled chaotic systems

A chaotic attractor containing unstable periodic orbits with different numbers of unstable directions is said to exhibit unstable dimension variability (UDV). We present general mechanisms for the progressive development of UDV in uni- and bidirectionally coupled systems of chaotic elements. Our results are applicable to systems of dissimilar elements without invariant manifolds. We also quantify the severity of UDV to identify coupling ranges where the shadowability and modelability of such systems are significantly compromised.

Unstable dimension variability and synchronization of chaotic systems

The nonhyperbolic structure of synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits an unstable dimension variability, which is an extreme form of nonhyperbolicity. We analyze the dynamics in the synchronization manifold and in its transversal direction, where a tonguelike structure is formed, through a system of two coupled chaotic maps. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values. We also point out that unstable dimension variability is a cause of severe modeling difficulty.