Quasiperiodic Forcing Research Articles

MEL'NIKOV'S APPROXIMATION DOMINANCE: SOME EXAMPLES

We continue a previous paper to show that Mel'nikov's first order formula for part of the separatrix splitting of a pendulum under fast quasi periodic forcing holds, in special examples, as an asymptotic formula in the forcing rapidity.

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

A simple example is considered of Hill's equation $$\ddot x + (a^2 + bp(t))x = 0$$ , where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

Regular and chaotic forced vibration of thin rotating rings

Dynamics of thin rings rotating with constant spin speed is investigated. The rings rotate with respect to their axis of rotational symmetry and they are under the influence of external forcing which leads to conditions of two-frequency or single-frequency primary resonance. First, a two-degree-of-freedom weakly non-linear model is employed, governing the amplitude of two in-plane bending modes of the ring with the same wave number. Then, a perturbation procedure is applied and modulation equations are derived for the amplitudes and phases of approximate analytical solutions of this model. For two-frequency quasiperiodic forcing, constant solutions of the modulation equations are shown to correspond to structural response involving combinations of forward and backward traveling waves, while for single-frequency resonant forcing the ring response is dominated by a forward or backward traveling wave. Characteristic effects of system parameters on the ring dynamics are then illustrated by representative series of response diagrams. Finally, direct integration of the modulation equations reveals rich dynamic behavior, involving coexistence and interaction of constant solutions with periodic and chaotic solutions.

Asymptotic behavior of solutions of 1D-Burgers equation with quasi-periodic forcing

Asymptotic behavior of solutions of 1D-Burgers equation with quasi-periodic forcing

Forced systems with almost periodic and quasiperiodic forcing term

Forced systems with almost periodic and quasiperiodic forcing term

Phase-locking in quasiperiodically forced systems

We study the influence of a quasiperiodic force on the phase-locking structure in the circle map. We show that with increasing forcing the phase-locking regions change their shape from a tongue with monotonically increasing width to the shape of a leaf where the width first grows and then shrinks to values close to zero. The change in the shape is accompanied by a change of the bifurcation type along the boundary curves of the phase-locking regions. For small forcing there is a transition from two- to three-frequency motion, but for larger forcing the transition is from two-frequency quasiperiodicity to a strange nonchaotic attractor.

Invariant graphs for forced systems

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When such contraction is uniform, it is well known that there exists a globally attracting invariant set which is the graph of a function φ from the driving state space to the driven state space. If the contraction is sufficiently strong, then φ is smooth. We describe the theoretical framework for such results and discuss a number of applications to the filtering of time series, to synchronization and to quasiperiodically forced systems. We go on to give recent generalizations to the case of non-uniform contraction; that is contraction measured by Lyapunov exponent like quantities. We conclude with a number of new results for quasiperiodic forcing; a corollary of these is that a strange non-chaotic attractor for such a system cannot be the graph of a continuous function, nor roughly speaking can it have as open attracting neighbourhood; in other words its closure must contain some repelling orbits.

On dimensions of attractive sets of viscous fluids on a sphere under quasi-periodic forcing

Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each set is the vorticity equation (VE) quasi-periodic solution of the complex (2n + 1)-dimensional subspace Hn of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its path being an open spiral densely wound around a 2n-dimensional torus in Hn , equals to 2n. As the generalized Grashof numb G becomes small enough then the basin of attraction of such spiral solution is expanded from Hn to the entire VE phase space. It is shown that for given G, there exists an integer nG such that each spiral solution generated by a forcing of Hn with n ≥ nG is globally asymptotically stable. Thus, whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may, for a fixed G, become arbitrarily large as the degree n of the quasi-periodic polynomial forcing grows. Since the small scale quasi-periodic functions, unlike the stationary ones, more adequately depict the barotropic atmosphere forcing, this result is of meteorological interest and shows that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure.

Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing

Quasiperiodic perturbations with two frequencies ( 1 / ε , γ / ε ) (1/\varepsilon ,\gamma /\varepsilon ) of a pendulum are considered, where γ \gamma is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for ε \varepsilon small enough. The value of the splitting, that turns out to be O ( exp ⁡ ( − c o n s t / ε ) ) \mathrm {O} \left (\exp \left (-\mathrm {const}/\sqrt {\varepsilon }\right )\right ) , is correctly predicted by the Melnikov function.

An Analytic Approach to Torus Bifurcation in a Quasiperiodically Forced Duffing Oscillator

A weakly nonlinear quasiconservative Duffing oscillator under quasiperiodic forcing is studied with the help of an analytic expression for the complex Poincare mapping. This mapping is then used to analyze the quasiperiodic response of the oscillator and the different zones of various periodicity. This map plays the same significant role as the averages equations in the theory of a periodically forced Duffing oscillator.

Multiband strange nonchaotic attractors in quasiperiodically forced systems

We study the effect of quasiperiodic forcing on two-dimensional invertible maps. As basic models the Hénon and the ring maps are considered. We verify the existence of strange nonchaotic attractors (SNA) in these systems by two methods which are generalized to higher dimensions: via bifurcation analysis of the rational approximations, and by calculating the phase sensitivity. Analyzing these systems we especially find a new phenomenon: the appearance of strange nonchaotic attractors which consist of 2 n bands. Similar to the band-merging crisis in chaotic systems, such a 2 n band SNA can merge to a 2 n−1 band SNA.

Nonexistence of Bounded Solutions of One Dimensional Wave Equations with Quasiperiodic Forcing Terms

Nonexistence of Bounded Solutions of One Dimensional Wave Equations with Quasiperiodic Forcing Terms

Quasiperiodic Forcing of a Chemical Reaction: Experiments and Calculations

Quasiperiodic Forcing of a Chemical Reaction: Experiments and Calculations

Locating bifurcations in quasiperiodically forced systems

We describe a method for locating and following saddle-node bifurcations of invariant circles in quasiperiodically forced systems. This is based on making successive rational approximations to the quasiperiodic forcing, and computing the locations of saddle-node bifurcations of appropriate periodic orbits for the approximating syste. An example of the application of the algorithm to the quasiperiodically forced sine map is given. The method is also equally applicable to period doubling bifurcations of invariant circles.

Characterizing strange nonchaotic attractors.

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. (c) 1995 American Institute of Physics.

Existence of Stable Quasi-Periodic Solutions and Persistence of Business Fluctuations

This paper considers the differential equations system with quasi-periodic forcing functions and provides an existence theorem of a stable quasi-periodic solution for the system. The result is illustrated with a simple multisector business fluctuation model in the Keynesian tradition.

On turbulence and complex dynamic in a four-dimensional Peano-Hilbert space

The work outlines a fairly model-independent scenario for a dynamic system to possess strange nonchaotic behaviour in the presence of quasi-periodic forcing. Implications for ergodicity and turbulence are discussed in connection with Sierpenski and peano-Hilbert spaces.

DYNAMICAL RESPONSE OF THE QUASIPERIODICALLY FORCED FRANCK-FITZHUGH MODEL FOR THE ELECTRODISSOLUTION OF IRON IN SULFURIC ACID SOLUTIONS: OBSERVATION OF QUASIPERIODIC, STRANGE NONCHAOTIC AND CHAOTIC BEHAVIOR

The response of the two-frequency quasiperiodically forced Franck-FitzHugh model is investigated. The Franck-FitzHugh model is a two-dimensional generic model which describes electrochemical oscillations at the transition of iron between the active and passive state. State variables are the potential where passivity occurs, the so-called Flade potential, and the fraction of the surface coverage. By applying a two-frequency quasiperiodic forcing, the two-dimensional model becomes of a four-dimensional one and this is integrated numerically. The response to a quasiperiodic forcing is characterized by the existence of two and three frequency torus, strange nonchaotic and chaotic attractors. Special emphasis is given to understanding the observed dynamic behavior in two parameter spaces: (i) [Formula: see text] for constant values of the other parameters of the system, where [Formula: see text] and ω1 are the amplitude and one of the frequencies of the applied quasiperiodically forced potential, whereas ω0 is the natural frequency of the autonomous system, and (ii) [Formula: see text] where f is a parameter of the system. Several methods of distinguishing the various types of attractors are used, such as Lyapunov exponents, surface of section plots, winding numbers and power spectra. A possible route to chaos is suggested for the forced Franck-FitzHugh model, considered in the present study, through the sequence: two-frequency quasiperiodicity → three-frequency quasiperiodicity → chaos.

Complex dynamics in a 4D Peano-Hilbert space

The work outlines a scenario for a dynamic system to possess strange nonchaotic behaviour in the presence of quasi-periodic forcing. Implications for ergodicity and turbulence are also discussed in connection with Serpenski and Peano-Hilbert spaces in 4D. Relation to quantum space-time is also touched upon.

Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation.

The dynamics associated with a quasiperiodically forced Morse oscillator is studied as a classical model for molecular dissociation under external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase-space barriers, allowing escape from bounded to unbounded motion. In contrast to the ubiquitous Poincar\'e map reduction of a periodically forced system, we derive a sequence of nonautonomous maps from the quasiperiodically forced system. We obtain a global picture of the dynamics, i.e., of transport in phase space, using a sequence of time-dependent two-dimensional lobe structures derived from the invariant homoclinic tangle of a persisting invariant saddle-type torus in a Poincar\'e section of an associated autonomous system phase space. Transport is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, and this provides the framework for studying basic features of molecular dissociation in the context of classical phase-space trajectories. We obtain a precise criterion for discerning between bounded and unbounded motion in the context of the forced problem. We identify and measure analytically the flux associated with the transition between bounded and unbounded motion, and study dissociation rates for a variety of initial phase-space ensembles, such as an even or weighted distribution of points in phase space, or a distribution on a particular level set of the unperturbed Hamiltonian (corresponding to a quantum state). A double-phase-slice sampling method allows exact numerical computation of dissociation rates. We compare single- and two-frequency forcing. Infinite-time average flux is maximal in a particular single-frequency limit; however, lobe penetration of the level sets of the unperturbed Hamiltonian can be maximal in the two-frequency case. The variation of lobe areas in the two-frequency problem gives one added freedom to enhance or diminish aspects of phase-space transport on finite time scales for a fixed infinite-time average flux, and for both types of forcing the geometry of lobes is relevant. The chaotic nature of the dynamics is understood in terms of a traveling horseshoe map sequence.