Creation Of Strange Nonchaotic Attractors Research Articles

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation

A new route to strange nonchaotic attractor (SNA), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series LCR circuit with a simple nonlinear element. Upon increasing the system control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighborhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincaré map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behavior via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are characterized by Poincaré maps. The multilayered bubble route to SNA is then confirmed by experimental Poincaré maps and explicit analytical solution is developed to further confirm it. Numerically observed bubbling route is characterized qualitatively in terms of phase portraits, power spectrum and further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with a simple nonlinear element

A new route to strange nonchaotic attractors (SNAs), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series $$\textit{LCR}$$ circuit with a simple nonlinear element. Upon increasing the control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighbourhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincare map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behaviour via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are charaterized by Poincare maps. Apart from the multilayered bubble route to SNA, Heagy–Hammel and fractalization routes to SNA have also been numerically observed and are characterized qualitatively interms of phase portraits, power spectrum. The above mentioned three routes are further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

Reliable Computation of Robust Response Tori on the Verge of Breakdown

We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy–Hammel route for the creation of strange nonchaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computer-assisted and are based on a tailored version of the Newton–Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques.

The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations

The author proposes a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which the author calls 'exponential evolution of peaks'.

Scenarios for generalized synchronization with chaotic driving

In chaotically driven nonlinear dynamical systems, weak generalized synchrony can arise through distinct scenarios or routes in a manner similar to the onset of low-dimensional chaos or the creation of strange nonchaotic attractors in quasiperiodically driven systems. The limit sets of the dynamics for weak generalized synchronization are nonchaotic-the Lyapunov exponent is nonpositive-and are geometrically strange. Quantitative measures related to the parameter sensitivity exponent and finite-time Lyapunov exponent distributions can be defined in order to characterize generalized synchronization.

APERIODIC NONCHAOTIC ATTRACTORS, STRANGE AND OTHERWISE

The main conceptual issues in the study of strange nonchaotic dynamics have been summarized and reviewed earlier [Prasad et al., 2001]. In the past five years, there has been further progress in the analysis of such attractors, and in understanding the nature of dynamical transitions in quasiperiodically forced systems. Here we discuss new developments which include the creation of strange nonchaotic attractors in excitable systems, the elucidation of the mechanisms for the intermittent and fractalization routes to SNA as well as their potential applications. The possibility of creating such attractors without recourse to quasiperiodic forcing is an important objective, and this has been realized in specific circumstances.

Strange nonchaotic attractors in the externally modulated Rayleigh–Bénard system

The amplitude equation associated with an externally modulated Rayleigh–Bénard system of binary mixtures near the codimension-two point is considered. Strange nonchaotic dynamics and chaotic behaviour are investigated numerically. The creation of strange nonchaotic attractors as well as the onset of chaos are studied through an analysis of Poincaré surfaces, a construction of the bifurcation diagram and a new method for computing Lyapunov exponents that exploits the underlying symplectic structure of Hamiltonian dynamics [Phys. Rev. Lett. 74 (1995) 70].

STRANGE NONCHAOTIC ATTRACTOR IN HIGH-DIMENSIONAL NEURAL SYSTEM

A general method for the creation of strange nonchaotic attractor is proposed. As an example, the strange nonchaotic attractor in a high-dimensional globally coupled neural system is studied numerically. For such an attractor, the time interval of continuously positive finite-time Lyapunov exponent must be smaller than the period of the driving stimulus, although it has a long-time positive tail. The intermittency between laminar and burst behavior is a characteristic dynamic of the strange nonchaotic attractors. Simulation results show that the chaotic phase occurs only within a small region around the origin in the parameter space. More than half of the large nonchaotic region is the strange nonchaotic phase. The chaotic phase is typically surrounded by strange nonchaotic attractors. This result also suggests that some biological signals that have a strange structure may be nonchaotic rather than chaotic.

Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven duffing oscillator

Different mechanisms for the creation of strange nonchaotic attractors (SNAs) are studied in a two-frequency parametrically driven Duffing oscillator. We focus on intermittency transitions in particular, and show that SNAs in this system are created through quasiperiodic saddle-node bifurcations (type-I intermittency) as well as through a quasiperiodic subharmonic bifurcation (type-III intermittency). The intermittent attractors are characterized via a number of Lyapunov measures including the behavior of the largest nontrivial Lyapunov exponent and its variance, as well as through distributions of finite-time Lyapunov exponents. These attractors are ubiquitous in quasiperiodically driven systems; the regions of occurrence of various SNAs are identified in a phase diagram of the Duffing system.

Collision and Symmetry Breaking in the Transition to Strange Nonchaotic Attractors

Strange nonchaotic attractors (SNAs) can be created due to the collision of an invariant curve with itself. This novel homoclinic transition to SNAs occurs in quasiperiodically driven maps which derive from the discrete Schrodinger equation for a particle in a quasiperiodic potential. In the classical dynamics, there is a transition from torus attractors to SNAs, which, in the quantum system, is manifest as the localization transition. This equivalence provides new insight into a variety of properties of SNAs, including its fractal measure. Further, there is a symmetry breaking associated with the creation of SNAs which rigorously shows that the Lyapunov exponent is nonpositive. We show that these characteristics associated with the appearance of SNA are robust and occur in a large class of systems.