Bifurcation structures of periodically forced oscillators

A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned.     

Ergodicity breaking induced by external coupled spatial-time noise processes

We report on the dynamics in a parameter plane of a continuous-time damped system driven by a periodic forcing. The dynamics is characterized by considering the Lyapunov exponents spectrum and conventional bifurcation diagrams, to discriminate periodic, quasiperiodic, and chaotic behaviors for each point in this parameter plane, according two parameters are simultaneously varied. Periodic structures born in a quasiperiodic region and embedded in a chaotic region, the so-called Arnold tongues, are observed. We show that the Arnold tongues periodic distribution is highly organized in a mixed set of two period-adding sequences. Other three typical periodic structures born and embedded in a chaotic region were observed, also individually organized in a mixed set of two period-adding sequences.     

Oscillations, period doublings, and chaos in CO oxidation and catalytic mufflers

Early experimental observations of chaotic behavior arising via the period-doubling route for the CO catalytic oxidation both on Pt(110) and Ptgamma-Al(2)O(3) porous catalyst were reported more than 15 years ago. Recently, a detailed kinetic reaction scheme including over 20 reaction steps was proposed for the catalytic CO oxidation, NO(x) reduction, and hydrocarbon oxidation taking place in a three-way catalyst (TWC) converter, the most common reactor for detoxification of automobile exhaust gases. This reactor is typically operated with periodic variation of inlet oxygen concentration. For an unforced lumped model, we report results of the stoichiometric network analysis of a CO reaction subnetwork determining feedback loops, which cause the oscillations within certain regions of parameters in bifurcation diagrams constructed by numerical continuation techniques. For a forced system, numerical simulations of the CO oxidation reveal the existence of a period-doubling route to chaos. The dependence of the rotation number on the amplitude and period of forcing shows a typical bifurcation structure of Arnold tongues ordered according to Farey sequences, and positive Lyapunov exponents for sufficiently large forcing amplitudes indicate the presence of chaotic dynamics. Multiple periodic and aperiodic time courses of outlet concentrations were also found in simulations using the lumped model with the full TWC kinetics. Numerical solutions of the distributed model in two geometric coordinates with the CO oxidation subnetwork consisting of several tens of nonlinear partial differential equations show oscillations of the outlet reactor concentrations and, in the presence of forcing, multiple periodic and aperiodic oscillations. Spatiotemporal concentration patterns illustrate the complexity of processes within the reactor.     

Phase synchronization in tilted deterministic ratchets

We study phase synchronization for a ratchet system. We consider the deterministic dynamics of a particle in a tilted ratchet potential with an external periodic forcing, in the overdamped case. The ratchet potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of external periodic forcing. This oscillator has an intrinsic frequency that can be entrained with the frequency of the external driving. We introduced a linear phase through a set of discrete time events and the associated average frequency, and show that this frequency can be synchronized with the frequency of the external driving. In this way, we can properly characterize the phenomenon of synchronization through Arnold tongues, which represent regions of synchronization in parameter space, and discuss their implications for transport in ratchets.     

Arnold tongues in a microfluidic drop emitter

The Letter reports an experimental study of microfluidic droplets produced in T junctions and subjected to a local periodic forcing. Synchronized and quasiperiodic regimes--organized into Arnold tongues and devil staircases--are reported for the first time for a system dedicated to drop emission. The nature of the dynamical regime controls the droplet characteristics. These phenomena are mostly controlled by the characteristics of the forcing and the flow conditions.     

Self-similarities of periodic structures for a discrete model of a two-gene system

We report self-similar properties of periodic structures remarkably organized in the two-parameter space for a two-gene system, described by two-dimensional symmetric map. The map consists of difference equations derived from the chemical reactions for gene expression and regulation. We characterize the system by using Lyapunov exponents and isoperiodic diagrams identifying periodic windows, denominated Arnold tongues and shrimp-shaped structures. Period-adding sequences are observed for both periodic windows. We also identify Fibonacci-type series and Golden ratio for Arnold tongues, and period multiple-of-three windows for shrimps.     

Cardiac arrhythmias and circle maps-A classical problem

The periodic forcing of nonlinear oscillations can often be cast as a problem involving self-maps of the circle. Consideration of the effects of changes in the frequency and amplitude of the periodic forcing leads to a problem involving the bifurcations of circle maps in a two-dimensional parameter space. The global bifurcations in this two-dimensional parameter space is described for periodic forcing of several simple theoretical models of nonlinear oscillations. As was originally recognized by Arnold, one motivation for the formulation of these models is their connection with theoretical models of cardiac arrhythmias originating from the competition and interaction between two pacemakers for the control of the heart.     

Arnold tongues in human cardiorespiratory systems

Arnold tongues are phase-locking regions in parameter space, originally studied in circle-map models of cardiac arrhythmias. They show where a periodic system responds by synchronizing to an external stimulus. Clinical studies of resting or anesthetized patients exhibit synchronization between heart-beats and respiration. Here we show that these results are successfully modeled by a circle-map, neatly combining the phenomena of respiratory sinus arrhythmia (RSA, where inspiration modulates heart-rate) and cardioventilatory coupling (CVC, where the heart is a pacemaker for respiration). Examination of the Arnold tongues reveals that while RSA can cause synchronization, the strongest mechanism for synchronization is CVC, so that the heart is acting as a pacemaker for respiration.     

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also presents multistability shown in the planes of the basin of attractions.     

Evidence of holes in the Arnold tongues of flow past two oscillating cylinders

The wake of two oscillating cylinders in a tandem arrangement is a nonlinear system that displays Arnold tongues. We show by numerical simulations that their geometry depends on the phase difference theta between the two oscillating cylinders. At theta = 0 there may be holes inside these intraresonance regions unlike the solid Arnold tongues encountered in single-cylinder oscillations. This implies that, surprisingly, self-excitation of the system may be suppressed inside these holes, at conditions close to its natural frequency.     

Resonance in Defect Turbulence under Periodic External Force

The periodically forced spatially extended Brusselator is investigated in the chaotic regime. We explore resonant or non-resonant patterns generated under various forcing frequencies and forcing amplitudes. Resonant spatially uniform oscillation and irregular structures are found. Furthermore two types of regular spatial patterns are generated under appropriate parameters. Our results of numerical simulations demonstrate that periodic force can give rise to resonant patterns in forced systems of spatiotemporal chaos similar to the situation of forced systems of regular oscillations.     

Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning

Dynamical system under investigation in the current work is comprised of harmonically forced linear oscillator with attached nonlinear energy sink. External forcing frequency detuning near the main resonance (1:1) is included in the system investigation. The detailed study of the periodic and quasiperiodic regimes is done in the work via (adaptive) averaging method. Local bifurcations of the periodic regimes are revealed and fully described in the space of system parameters (amplitude of excitation, damping, and frequency detuning). Novel analytical approach for predictions of strongly modulated response (SMR) is presented. This approach provides a sufficient condition for the SMR existence contrary to the previous studies. Various possibilities of coexistence of the response regimes are predicted analytically and demonstrated numerically. Among those is a coexistence of two distinct periodic regimes together with the SMR. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.     

Lock-in and quasiperiodicity in hydrodynamically self-excited flames: Experiments and modelling

Hydrodynamically self-excited flames are often assumed to be insensitive to low-amplitude external forcing. To test this assumption, we apply acoustic forcing to a range of jet diffusion flames. These flames have regions of absolute instability at their base and this causes them to oscillate at discrete natural frequencies. We apply the forcing around these frequencies, at varying amplitudes, and measure the response leading up to lock-in. We then model the system as a forced van der Pol oscillator.Our results show that, contrary to some expectations, a hydrodynamically self-excited flame oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, it responds at both frequencies as well as at several nearby frequencies, indicating quasiperiodicity. When forced at high amplitudes, it locks into the forcing. The critical forcing amplitude for lock-in increases both with the strength of the self-excited instability and with the deviation of the forcing frequency from the natural frequency. Qualitatively, these features are accurately predicted by the forced van der Pol oscillator. There are, nevertheless, two features that are not predicted, both concerning the asymmetries of lock-in. When forced below its natural frequency, the flame is more resistant to lock-in, and its oscillations at lock-in are stronger than those of the unforced flame. When forced above its natural frequency, the flame is less resistant to lock-in, and its oscillations at lock-in are weaker than those of the unforced flame. This last finding suggests that, for thermoacoustic systems, lock-in may not be as detrimental as it is thought to be.     

双驱动布鲁塞尔振子的一些运动性质

以强迫布鲁塞尔振子方程为例,研究了外加第二驱动对振子运动性质的影响:某些有理频率比的第二驱动可以抑制方程的混沌运动,得到周期轨道;而某些无理频率比的外驱动则可以改变方程的运动性质,造成非混沌的奇怪吸引子。     

Control and managing of localized states in two-dimensional systems with periodic forcing

We study the formation of localized structures in two-dimensional systems with periodic forcing, showing that these types of systems provide an adequate framework for the study and control of localized structures. Theoretically, we introduce a dissipative ϕ ⁴ model as a prototype for a bistable spatially forced system, and we show that with different spatial forcings of small amplitudes, such as square or hexagonal grids, this model exhibits a family of localized structures. By changing the forcing parameters, we control the bistability between the various induced patterns. Experimentally, based on an optical feedback with spatially amplitude-modulated beam, we set-up a two-dimensional forced experiment in a nematic liquid crystal cell. By changing the forcing parameters, the system exhibits a family of localized structures that are confirmed by numerical simulations for the average liquid crystal tilt angle.     

Resonantly forced inhomogeneous reaction-diffusion systems

The dynamics of spatiotemporal patterns in oscillatory reaction-diffusion systems subject to periodic forcing with a spatially random forcing amplitude field are investigated. Quenched disorder is studied using the resonantly forced complex Ginzburg-Landau equation in the 3:1 resonance regime. Front roughening and spontaneous nucleation of target patterns are observed and characterized. Time dependent spatially varying forcing fields are studied in the 3:1 forced FitzHugh-Nagumo system. The periodic variation of the spatially random forcing amplitude breaks the symmetry among the three quasi-homogeneous states of the system, making the three types of fronts separating phases inequivalent. The resulting inequality in the front velocities leads to the formation of "compound fronts" with velocities lying between those of the individual component fronts, and "pulses" which are analogous structures arising from the combination of three fronts. Spiral wave dynamics is studied in systems with compound fronts. (c) 2000 American Institute of Physics.     

Bistability in Coupled Oscillators Exhibiting Synchronized Dynamics

We report some new results associated with the synchronization behavior of two coupled double-well Duffing oscillators （DDOs）. Some sufficient algebraic criteria for global chaos synchronization of the drive and response DDOs via linear state error feedback control are obtained by means of Lyapunov stability theory. The synchronization is achieved through a bistable state in which a periodic attractor co-exists with a chaotic attractor. Using the linear perturbation analysis, the prevalence of attractors in parameter space and the associated bifurcations are examined. Subcritical and supercritical Hopf bifurcations and abundance of Arnold tongues -- a signature of mode locking phenomenon are found.     

Discrete coherent amplification of oscillations by nonresonant forcing

A new mechanism of generation of oscillations in a linear forced oscillatory system is found. Natural oscillations may be generated at a "sharp" pulse (rapid variation) of the natural frequency. In this process oscillations are generated by nonresonant forcing, e.g., by the action of a constant, nonperiodic or periodic force (with driving frequency much less than the natural one). Repetitive pulses of the natural frequency result in emergence of oscillations that interfere and may give a powerful resultant output. These phenomena relate to a basis of the theory of open linear oscillatory systems.     

Bistability,basins of attraction and predictability in a forced mass-reaction model

A simple mass-reaction model with periodically forced contact rate is considered. It is shown that the forced differential equation exhibits at least two distinct periodic orbits for several ranges of forcing amplitude. Basins of attraction are computed for co-existing stable periodic orbits and are shown to be intertwined in a complex manner. Dimension (capacity) of the basin is computed, and it is shown how the basin structure affects final state predictability.