Three-frequency Quasiperiodicity Research Articles

Dynamics of non–identical coupled Chialvo neuron maps

We study numerically the dynamics of low–dimensional ensembles of discrete neuron models - Chialvo maps. We are focused on choosing the autonomous map parameters corresponding to the invariant curve. We consider two cases of coupling organization: (i) via a nonlinear function of models; (ii) linear coupling, which is an analog of electrical neuron interaction. For the first case, the possibility of invariant curve doublings and the emergence of quasi-periodicity within Arnold tongues as a result of the secondary Neimark-Sacker bifurcation are found. For the second case, we discover an area of three-frequency quasi-periodicity for the case of two neurons. It arises softly as a result of quasi-periodic Hopf bifurcation. We demonstrate a set of resonant two-frequency regimes tongues embedded in this area and bounded by lines of saddle-node bifurcations of invariant curves. For ensemble of three linearly coupled maps, four-frequency quasi-periodicity becomes possible with a built-in system of tongues of three-frequency regimes (tori). We discuss the effect of noise and the evolution of “noise quasi-periodic” regimes, resonant regimes of this type and bifurcations of invariant tori with increasing of noise intensity.

Coupled systems with quasi-periodic and chaotic dynamics

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic Rössler system is studied. We have shown that with the growth of the coupling, regimes of two-frequency and three-frequency quasiperiodicity, a periodic regime and a regime of oscillation death sequentially arise. With a small coupling strength, doubling bifurcations of three-frequency tori are observed in the system. A chaotic regime, characterized by two additional zero Lyapunov exponents in spectrum, is revealed. Two-parameter Lyapunov exponent analysis and bifurcation analysis are presented. A new bifurcation scenario of transition from the regime of oscillation death to quasi-periodicity in coupled systems is described.

Chimera states in a can-annular combustion system

We experimentally investigate the collective behaviour of four turbulent lean-premixed combustors coupled in a ring network. We adopt a complex systems approach, treating each combustor as an individual self-excited thermoacoustic oscillator and examining how a minimal network of such oscillators can self-organize into synchronous and asynchronous states. Using proven synchronization metrics, we identify a broad range of collective behaviour, including a weak anti-phase chimera, a breathing chimera, intermittent frequency locking, and three-frequency quasiperiodicity. The discovery of chimera states in a minimal network of coupled thermoacoustic oscillators suggests that chimera control techniques could be exploited for the suppression of thermoacoustic oscillations in can-annular combustion systems.

О влиянии шума на квазипериодичность разной размерности, включая квазипериодическую бифуркацию Хопфа

Background and Objectives: The basic model of study is the simplest three - dimensional map with two-frequency and three-frequency quasiperiodicity at adding of noise. The main objective is to examine the effect of noise on the quasiperiodic Hopf bifurcation of the 3-torus birth. Materials and Methods: To study the torus map in the presence of noise we use such numerical methods as computing of Lyapunov exponents, calculation of Fourier spectra, drawing of attractor portraits. Results: Quasi-periodic bifurcations under the influence of noise occupy certain intervals in the parameter, but their main classification features (equality or not of the corresponding Lyapunov exponents) are preserved at the qualitative level. Conclusion: We considered the effect of noise on the simplest system with two- and three-frequency quasiperiodicity. The three-frequency quasiperiodicity is preserved at certain noise amplitudes, but then turns into a two-frequency one. In the Fourier spectra, this process develops according to the scenario of "blurring" the noise components of the corresponding spectral components.

The Simplest Map with Three-Frequency Quasi-Periodicity and Quasi-Periodic Bifurcations

We propose a new three-dimensional map that demonstrates the two- and three-frequency quasi-periodicity. For this map, all basic quasi-periodic bifurcations are possible. The study was realized by using Lyapunov charts completed by plots of Lyapunov exponents, phase portraits and bifurcation trees illustrating the quasi-periodic bifurcations. The features of the three-parameter structure associated with quasi-periodic Hopf bifurcation are discussed. The comparison with nonautonomous model is carried out.

Synchronisation in the phase model of three coupled lasers

The problem of synchronisation of three lasers is considered within the phase approximation. The domains of complete synchronisation, partial synchronisation, two-frequency resonant regimes, and three-frequency quasi-periodicity have been found using bifurcation analysis, the method of Lyapunov exponent maps, and construction of phase portraits. The differences in the properties of a three-element chain and ring, as well as the influence of the coupling type, are discussed.

Complex Interaction Between Tori and Onset of Three-Frequency Quasi-Periodicity in a Current Mode Controlled Boost Converter

It is known that power electronic circuits like dc-dc converters are highly nonlinear systems, and that period doubling and Neimark-Sacker bifurcations are common sources of instability in such systems. It has also been shown that these two types of bifurcation may interact, giving rise to interesting dynamical phenomena. In this paper we show that in a current mode controlled dc-dc converter, periodic, quasi-periodic, and saturation behavior can coexist for the same parameter value, and there can be complex interactions between them. Furthermore, abrupt exit to saturation mode can be triggered by a torus-torus collision. Finally, we report the first observation of three-frequency quasi-periodicity in a power electronic system.

Low-frequency behavior in a frictionally excited beam

A cantilevered beam excited by a periodically reciprocating friction contact surface exhibited extremely low-frequency responses (with frequencies as low as 100 times lower than the driver). Example responses near 1:1 and 1:2 resonances, between the excitation and the degree of freedom normal to the direction of sliding, show two-frequency quasiperiodicity, and in one case three-frequency quasiperiodicity. Underlying circle maps were extracted, and winding numbers were matched to response frequencies quantified in the fast Fourier transforms of the responses. A torus doubling bifurcation was documented.

Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map

We consider the dynamics of small perturbations of stable two-frequency quasiperiodic orbits on an attracting torus in the quasiperiodically forced Henon map. Such dynamics consists in an exponential decay of the radial component and in a complex behaviour of the angle component. This behaviour may be two- or three-frequency quasiperiodicity, or it may be irregular. In the latter case a graphic image of the dynamics of the perturbation angle is a fractal object, namely a strange nonchaotic attractor, which appears in auxiliary map for the angle component. Therefore, we claim that stable trajectories may approach the attracting torus either in a regular or in an irregular way. We show that the transition from quasiperiodic dynamics to chaos in the model system is preceded by the appearance of an irregular behaviour in the approach of the perturbed quasiperiodic trajectories to the smooth attracting torus. We also demonstrate a link between the evolution operator of the perturbation angle and a quasiperiodically forced circle mapping of a special form and with a Harper equation with quasiperiodic potential.

PHASE-RESETTING MAP AND THE DYNAMICS OF QUASI-PERIODICALLY FORCED BIOLOGICAL OSCILLATORS

In this paper we study the dynamics of a biological oscillator forced at two incommensurate frequencies which, as our results show, can exhibit a variety of novel dynamical phenomena, including two-frequency quasi-periodicity, three-frequency quasi-periodicity, chaos and, in particular, strange nonchaotic attractors. We discuss the significance and observable signatures of these phenomena, and suggest biological systems in which these phenomena may be experimentally investigated.

Observation of chaotic instability in the active mode locking of a semiconductor laser

We examine experimentally the consequence of frequency detuning an actively mode-locked external-cavity semiconductor laser from resonance. We observe a transition of the laser system from a periodic oscillation to a nonperiodic state with broadened spectral tones. By estimating the fractal dimension of the corresponding phase-space attractors, we show the presence of low-dimensional chaos. The route to chaos is a well-defined regime of three-frequency quasi-periodicity preceded by a two-frequency quasi-periodicity.

Three-frequency quasiperiodicity and chaos in a neon glow discharge

We present detailed experimental studies of regular and chaotic motion in a periodically perturbed neon glow discharge. Due to the simultaneous presence of two competing fundamental oscillatory modes intrinsic to our discharge system and their nonlinear interaction with the external periodic forcing two- and three-frequency quasiperiodic and mode locked states as well as low-dimensional chaos can be observed. In our experiment, chaos is distinguished from quasiperiodicity by evaluation of the temporal decay of the envelope of the time series autocorrelation function and careful estimation of the correlation dimension of the corresponding attractor.

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.

Three-frequency quasiperiodicity, phase locking, and the onset of chaos

We have performed a series of experiments with coupled relaxation oscillators to study three-frequency quasiperiodicity. There is strong evidence for the Ruelle-Takens scenario for the transition to chaos although only a very small portion of parameter space is occupied by chaos. The chaotic transition is via an exchange of stability between states which have phase locked to two-frequency quasiperiodicity. The overall structure of parameter space is a very complicated fractal devil's cobweb of periodic resonances linked by bands of two-frequency phase lockings.

Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic.

As a model displaying typical features of two-frequency quasiperiodically forced systems, we discuss the circle map with quasiperiodic coupling. We present numerical and analytical evidence for the existence of strange nonchaotic attractors, and we use examples to illustrate various types of dynamical behavior that can arise in typical quasiperiodically forced systems. We investigate the behavior of the system in the two-dimensional parameter plane of nonlinearity strength versus one of the driving frequencies. We find that the set in this parameter plane for which the system exhibits strange nonchaotic attractors has Cantor-like structure and is enclosed between two critical curves. One of these curves marks the transition from three-frequency quasiperiodic attractors to strange nonchaotic attractors; the other marks the transition from strange nonchaotic attractors to chaotic attractors. This suggests a possible route to chaos in two-frequency quasiperiodically forced systems: (three-frequency quasiperiodicity)..-->..(strange nonchaotic behavior)..-->..(chaos).

THREE-FREQUENCY QUASIPERIODIC TRANSITION TO CHAOS IN OPTICAL BISTABILITY

The dynamic behavior of a hybrid optical bistability with two delayed feedback loops under the adiabatic approximation condition is studied. In this models, the system can be described by high dimensional mappings. Increasing input intensity while keeping the ratio of two time delays fixed at ω = 3/5, we observe transitions between three-frequency quasiperio-dicity and chaos. This result could be of importance in our understanding of turbulence.

Persistence of three-frequency quasiperiodicity under large perturbations.

Persistence of three-frequency quasiperiodicity under large perturbations.

Three-frequency quasiperiodicity and chaos on a torus

Three-frequency quasiperiodicity and chaos on a torus

Quasiperiodic transitions to chaos of instabilities in an electron-hole plasma excited by ac perturbations at one and at two frequencies.

Periodic instabilities of frequency f/sub 0/ in an electron-hole plasma in Ge are excited with an external perturbation V/sub 1/ sin(2..pi..f/sub 1/t). Varying V/sub 1/ and f/sub 1/ results in quasiperiodic, mode-locked, and chaotic states comparable to those expected theoretically. Increasing V/sub 1/ while keeping the ratio f/sub 0//f/sub 1/ fixed at the golden mean (1.6180 . . .), we observe a universal transition to chaos, in quantitative agreement with theory. Driving the instability simultaneously at two incommensurate frequencies, f/sub 1/ and f/sub 2/, we observe transitions between three-frequency quasiperiodicity, two-frequency mode locking, and chaos.