Quasi-periodic Hopf Bifurcation Research Articles

Discrete Rössler Oscillators: Maps and Their Ensembles

We study the complex dynamics of a discrete analogue of the classical flow dynamical system — Rössler oscillator. Minimal ensembles of two and three coupled discrete oscillators with different topologies are considered. As the main research tool we used the method of Lyapunov exponents charts. For coupled systems, the possibility of two-, three- and four-frequency quasi-periodicity is revealed. Illustrations in the form of Fourier spectra are presented. Doublings of invariant curves, two- and three-dimensional tori are found. The transition from two-dimensional tori to three-dimensional ones occurs through a quasi-periodic saddle-node bifurcation of invariant tori or through a quasi-periodic Hopf bifurcation. A discrete version of the hyperchaotic Rössler oscillator is also discussed. It exhibits dynamical behavior close to a flow system in some measure.

Hopf and quasi-periodic Hopf bifurcations and deterministic coherence in coupled Duffing-Holmes and Van der Pol oscillators: the Arnol’d resonance web

A survey on coupled Van der Pol (VDP) and Duffing-Holmes (DH) oscillators, a model widely encountered in various branches of Physics and engineering is done. A bifurcation analysis of the model is presented when both unidirectional and mutual couplings are considered. It is found that stable, unstable and chaotic behaviors appear in the models. Using the method of charts of dynamics regimes in parameter planes, numerical study of the parameters space of the initial differential equations is done. Results from both approaches are compared and discussed. Features of the bifurcation picture are discussed when varying control parameters and analysis of slow-flow equations is presented. We show that the local bifurcation transition from an invariant one-torus (IT1) to an invariant two-torus (IT2) is caused by a Neimark-Saker (NS) bifurcation, also known as a one dimension-higher quasi-periodic Hopf (QH) bifurcation, these by analysing the graph of Lyapunov exponents. We observe that in the mutual coupling case, a complex structure generally-called Arnold’s resonance web phenomenon appears. Numerical simulations are compared to experimental measurements to illustrate the above behaviors. We also note the birth of coherence resonance in the slave oscillator for a certain coupling strength in the case of unidirectional coupling. We also analyze the effect of the coupling strength on the generalized synchronization between the master system and the slave system in the case of unidirectional coupling, through the concept of mutual false nearest neighbors. It allows us to understand when and how closeness in response space implies closeness in driving space.

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.

Evolution of quasiperiodicity in quorum-sensing coupled identical repressilators

The dynamics of three three-dimensional repressilators globally coupled by a quorum sensing mechanism was numerically studied. This number (three) of coupled repressilators is sufficient to obtain such a set of self-consistent oscillation frequencies of signal molecules in the mean field that results in the appearance of self-organized quasiperiodicity and its complex evolution over wide areas of model parameters. Numerically analyzing the invariant curves as a function of coupling strength, we observed torus doubling, three torus arising via quasiperiodic Hopf bifurcation, the emergence of resonant cycles, and secondary Neimark-Sacker bifurcation. A gradual increase in the oscillation amplitude leads to chaotizations of the tori and to the birth of weak, but multidimensional chaos.

Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena

Abstract A problem of synchronization of quasiperiodic oscillations is discussed in application to an example of coupled systems with autonomous quasiperiodic dynamics. Charts of Lyapunov exponents are presented that reveal characteristic domains on the parameter plane such as oscillator death, complete synchronization, phase synchronization of quasiperiodic oscillations, broadband synchronization, broadband quasiperiodicity. Features of each kind of dynamical behavior are discussed. Analysis of corresponding bifurcations is presented, including quasiperiodic Hopf bifurcations, saddle–node bifurcations of invariant tori of different dimensions, and bifurcations of torus doublings. Both the case of dominance of quasiperiodic oscillations in one of the generators and the case of pronounced periodic resonances embedded in the region of quasiperiodicity are considered.

Bifurcation and stability analysis of a nonlinear rotor system subjected to constant excitation and rub-impact

Abstract This paper focuses on the mechanism of a complex bifurcation behavior caused by flight maneuvers in an aircraft rub-impact rotor system with Duffing type nonlinearity. The maneuver load during flight maneuvers may induce a rub-impact phenomenon, accompanied by complex nonlinear behaviors such as periodic, sub-harmonic and quasi-periodic motions, but the bifurcation mechanism is not so clear. In this study, the harmonic balance method combined with an alternating frequency/time domain procedure (HB-AFT method) is formulated and used to derive the approximate periodic solutions of the system. In conjunction with the arc-length continuation, the solution branches for both periodic 1-T motion (synchronous oscillation) and periodic 2-T motion (sub-harmonic oscillation) are traced. Then with the aid of the Floquet theory, the stabilities of the obtained periodic solutions are examined. The changes in the number or the stability of the solutions lead to the identification of the bifurcation points, which can be classified qualitatively into three types, i.e. Neimark-Sacker bifurcation point (NSBP), quasi-periodic Hopf bifurcation point (QPHBP) and saddle-node bifurcation point (SNBP). In addition, the constant excitation significantly affects the instability as well as the bifurcation of the rotor system. In the case of a smaller constant excitation, the rotation region with respect to the quasi-periodic motions may disappear, accompanied with a PBP (pitchfork bifurcation point) instead of the two NSBPs and one QPHBP. The bifurcation analysis in this paper provides deep insight into the mechanism of the complex nonlinear phenomenon induced by the constant excitation. The results obtained will also contribute to a better understanding of the nonlinear dynamic behaviors of aircraft rotor systems during flight maneuvers.

Quasi-periodic bifurcations of four-frequency tori in the ring of five coupled van der Pol oscillators with different types of dissipative coupling

Five van der Pol oscillators connected into a ring have been investigated numerically. We have considered different types of coupling: dissipative and active, as well as dissipative and active with a single coupling sign reversal. Bifurcations observed upon a transition from a five- to four-frequency torus have been investigated. The possibility of quasi-periodic Hopf bifurcation in this system has been established.

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.

Numerical investigation of electrohydrodynamic instability and bifurcation in a dielectric liquid subjected to unipolar injection

The electroconvection of dielectric liquids subjected to unipolar injection is numerically studied in a two-dimensional cavity. To understand the Coulomb force driven electroconvective instabilities and bifurcations in the closed cavity, a high resolution upwind scheme is applied to carry out linear stability analysis and numerical simulation of nonlinear electrohydrodynamic equations. We focus on the strong injection case, where the non-dimensional injection parameter C is fixed at C=10. The length to height ratio of the cavity is fixed at 0.614. The numerical simulations are performed subject to free and rigid lateral wall boundary conditions. Two dimensionless mobility parameters M=5 and M=10 are considered for each kind of the lateral boundary conditions. Our linear stability analysis result is consistent with previous theoretical predictions. The nonlinear behaviors of the investigated system beyond the convection threshold are carefully examined. Abundant bifurcation phenomena, such as the pitchfork bifurcation, Hopf bifurcation, quasi-periodic Hopf bifurcation and heteroclinic bifurcation, have been observed. Some nonlinear flow features, such as the hysteresis loops, coexistence of multiple solutions and transition to chaos, have also been demonstrated. Our simulation results show that flow structure and bifurcation sequence can be greatly affected by the specified lateral boundary conditions and the chosen mobility parameters. Besides the static state, a series of steady and unsteady solutions has been calculated, like the one-cell steady state, one-cell periodic state, one-cell quasi-periodic state, and the one-cell chaotic state. All these convective states with two-cell structure are also obtained. Moreover, a periodic state oscillating between the one-cell structure and the two-cell structure has been found in flow domain bounded by rigid walls.

From chaos to quasi-periodicity

Ensembles of several Rossler chaotic oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of different and sufficiently high dimensional invariant tori. The possibility of a quasi-periodic Hopf bifurcation and a cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonance tori are revealed. Boundaries of these domains correspond to the saddle-node bifurcations. Inside the domains of resonance modes, torus-doubling bifurcations and destruction of tori are observed.

Bifurcation analysis of four-frequency quasi-periodic oscillations in a three-coupled delayed logistic map

We present herein an extensive analysis of the bifurcation structures of quasi-periodic oscillations generated by a three-coupled delayed logistic map. Oscillations generate an invariant three-torus, which corresponds to a four-dimensional torus in vector fields. We illustrate detailed two-parameter Lyapunov diagrams, which reveal a complex bifurcation structure called an Arnol'd resonance web. Our major concern in this study is to demonstrate that quasi-periodic saddle-node bifurcations from an invariant two-torus to an intermittent invariant three-torus occur because of a saddle-node bifurcation of a stable invariant two-torus and a saddle invariant two-torus. In addition, with some assumptions, we derive a bifurcation boundary between a stable invariant two-torus and a stable invariant three-torus due to a quasi-periodic Hopf bifurcation with a precision of 10−5.

Three-dimensional tori and Arnold tongues

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.

About Landau–Hopf scenario in a system of coupled self-oscillators

The conditions are discussed for which an ensemble of interacting oscillators may demonstrate the Landau-Hopf scenario of successive birth of multi-frequency quasi-periodic motions. A model is proposed that is a network of five globally coupled oscillators characterized by controlled degree of activation of individual oscillators. Illustrations are given for successive birth of tori of increasing dimension via quasi-periodic Hopf bifurcations.

The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web

A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.

Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

A low-dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three-dimensional Poincaré mapping depends on three control parameters F, G, and ϵ, the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of F,G,ϵ. For ϵ small, a Hopf-saddle-node bifurcation of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case ϵ = 0. For ϵ = 0.5, the above bifurcations have disappeared. Different types of strange attractors are found in four regions (chaotic ranges) in {F,G} and the related routes to chaos are discussed.

Resonances in skew and reducible quasi-periodic Hopf bifurcations

Two types of quasi-periodic Hopf bifurcations are known, in which a Whitney smooth family of quasi-periodic attractors loses its hyperbolicity. One is the reducible case, where the normal linear dynamics are trivial. Another is the skew case, where the normal dynamics are topologically non-trivial. There, the dynamics can involve periodicity, quasi-periodicity and chaos, including chaos with a mixed spectrum. This paper investigates a model system where the bifurcating circle supports Morse-Smale (also called resonant) dynamics. First this is done under the assumption of rotational symmetry (in the normal direction) of the system; a local bifurcation analysis is given for this. Then the effects of generic (non-symmetric) perturbations are discussed. It turns out that the bifurcation diagram is organized by several codimension-two bifurcations: saddle-Hopf and degenerate Hopf bifurcations for diffeomorphisms, and Bogdanov-Takens bifurcations of invariant circles. The (local) bifurcation diagram is computed analytically for the model system. We believe that this model is representative for the minimal dynamical complexity which generically occurs whenever the dynamics on the bifurcating circle is resonant.

On a quasi-periodic Hopf bifurcation

In this paper we study quasi-periodic Hopf bifurcations for the model problem of a quasi-periodically forced oscillator, where the frequencies remain fixed. For this purpose we first consider Stoker’s problem for small damping.