In the present survey, the contemporary state of the art of the problem of the justification of nonstationary linearizations is presented.

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Time-varying linearization and the perron effects

Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Second, we propose to characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown that phase sensitivity appears if there is a nonzero probability for positive local Lyapunov exponents to occur.

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Characterizing strange nonchaotic attractors.

Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrodinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.

Strange Nonchaotic Attractors

The complex dynamical behavior of a finance system is investigated in this paper The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated This research has important theoretical and practical meanings

Chaos and Hopf bifurcation of a finance system

Basics of Nonlinear and Chaotic Dynamics.- Phase Transitions and Synergetics.- Geometry and Topology Change in Complex Systems.- Nonlinear Dynamics of Path Integrals.- Complex Nonlinearity: Combining It All Together.

J. F. Heagy

Papers

The birth of strange nonchaotic attractors