Nonlinear dynamics investigation in parameter planes of a periodically forced compound KdV-Burgers equation

Parameter plane plots related to a periodically forced compound Korteweg-de Vries-Burgers system, which is modeled by a third-order partial differential equation, are reported. It is shown that typical periodic structures embedded in a chaotic region in these parameter planes, organize themselves in different ways. There are bifurcation sequences whose periods have a well-defined law of formation, that may be written in a closed form, and there are bifurcation sequences self-organized in period-adding cascades.     

Hyperchaos and quasiperiodicity from a four-dimensional system based on the Lorenz system

This paper reports on numerically computed parameter plane plots for a dynamical system modeled by a set of five-parameter, four autonomous first-order nonlinear ordinary differential equations. The dynamical behavior of each point, in each parameter plane, is characterized by Lyapunov exponents spectra. Each of these diagrams indicates parameter values for which hyperchaos, chaos, quasiperiodicity, and periodicity may be found. In fact, each diagram shows delimited regions where each of these behaviors happens. Moreover, it is shown that some of these parameter planes display organized periodic structures embedded in quasiperiodic and chaotic regions.     

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.     

Dynamics of a neuron model in different two-dimensional parameter-spaces

We report some two-dimensional parameter-space diagrams numerically obtained for the multi-parameter Hindmarsh-Rose neuron model. Several different parameter planes are considered, and we show that regardless of the combination of parameters, a typical scenario is preserved: for all choice of two parameters, the parameter-space presents a comb-shaped chaotic region immersed in a large periodic region. We also show that exist regions close these chaotic region, separated by the comb teeth, organized themselves in period-adding bifurcation cascades.     

Experimental evidence of a chaotic region in a neural pacemaker

In this Letter, we report the finding of period-adding scenarios with chaos in firing patterns, observed in biological experiments on a neural pacemaker, with fixed extra-cellular potassium concentration at different levels and taken extra-cellular calcium concentration as the bifurcation parameter. The experimental bifurcations in the two-dimensional parameter space demonstrate the existence of a chaotic region interwoven with the periodic region thereby forming a period-adding sequence with chaos. The behavior of the pacemaker in this region is qualitatively similar to that of the Hindmarsh–Rose neuron model in a well-known comb-shaped chaotic region in two-dimensional parameter spaces.     

一个不连续映象中的混沌稳定或混沌抑制

借助于一个张弛振子模型和与之相应的不连续映象说明了在这类系统中瞬态集的映蔽效应会产生三种区域：1）稳定混沌区，在此区域中不存在周期窗口，混沌轨道是结构稳定的；2）完全锁相区，在此区域中混沌被抑制，只存在周期运动；3）准周期区域，在此区域中混沌被抑制，只存在准周期或临界稳定的周期运动．这种思想被用来解释在一个实际张弛振子电路中观察到的稳定混沌区和完全锁相区 关键词：     

耦合电路中的复杂振荡行为分析

讨论了两个非线性电路适当连接后的耦合系统随耦合强度变化的演化过程.给出了两子系统各自的分岔行为及通向混沌的过程,指出原子系统均为周期运动时,耦合系统依然会由倍周期分岔进入混沌,同时在混沌区域中存在有周期急剧增加及周期增加分岔等现象.而当周期运动和混沌振荡相互作用时,在弱耦合条件下,受混沌子系统的影响,原周期子系统会在其原先的轨道邻域内作微幅振荡,其振荡幅值随耦合强度的增加而增大,混沌的特征越加明显,相反,周期子系统不仅可以导致混沌子系统的失稳,也会引起混沌吸引子结构的变化. 关键词： 非线性电路 耦合强度 分岔 混沌     

Alternate Phase Synchronization in Coupled Chaotic Oscillators

Phase locking dynamics in coupled chaotic oscillators is investigated.For chaotic systems with a poorly coherent phase variable,the imperfect phase locking can be observed befor the onset of a complete phase synchronization.The temporal alternations among n:n phase lockings are found,which originate from an overlap of m:n Arnold tongues.     

Shrimp-shape domains in a dissipative kicked rotator

Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime.     

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.     

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.     

Resonant and nonresonant patterns in forced oscillators

Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillator's frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.     

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction-diffusion lattice with complex threshold excitation

Space-time dynamics of the system modeling collective behaviour of electrically coupled nonlinear units is investigated. The dynamics of a local cell is described by the FitzHugh-Nagumo system with complex threshold excitation. It is shown that such a system supports formation of two distinct kinds of stable two-dimensional spatially localized moving structures without any external stabilizing actions. These are regular and polymorphic structures. The regular structures preserve their shape and velocity under propagation while the shape and velocity as well as other integral characteristics of polymorphic structures show rather complex temporal behaviour. Both kinds of structures represent novel sorts of spatially temporal patterns which have not been observed before in typical two-component reaction-diffusion type systems. It is demonstrated that there exist two types of regular structures: single and bound states and three types of polymorphic structures: periodic, quasiperiodic and even chaotic ones. The partition of the parameter plane into regions corresponding to the existence of these different types of structures is carried out. High multistability of regular structures is indicated. The interaction of regular structures is investigated. The correspondence between the structures and trajectories in multidimensional phase space associated with the system is given. Bifurcation mechanisms leading to the loss of stability of regular structures as well as to a transition from one type of polymorphic structure to another are indicated. The mechanisms of formation of regular and polymorphic structures are discussed.     

Dynamics of an erbium-doped fiber dual-ring laser

We report results of a numerical investigation on two-dimensional parameter-spaces of a set of four autonomous, seven-parameter, first-order ordinary differential equations, which models an erbium-doped fiber dual-ring laser. By using Lyapunov exponents to numerically characterize the dynamics of the model in the parameter-space, we show that it presents typical self-organized periodic structures embedded in a chaotic region.     

Torsion-adding and asymptotic winding number for periodic window sequences

In parameter space of nonlinear dynamical systems, windows of periodic states are aligned following the routes of period-adding configuring periodic window sequences. In state space of driven nonlinear oscillators, we determine the torsion associated with the periodic states and identify regions of uniform torsion in the window sequences. Moreover, we find that the measured torsion differs by a constant between successive windows in periodic window sequences. Finally, combining the torsion-adding phenomenon, reported in this work, and the known period-adding rule, we deduce a general rule to obtain the asymptotic winding number in the accumulation limit of such periodic window sequences.     

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.     

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis,chaos control and synchronisation in its fractional-order form

Dynamics at infinity and a Hopf bifurcation for a Sprott E system with a very small perturbation constant are studied in this paper. By using Poincaré compactification of polynomial vector fields in \(R^3\), the dynamics near infinity of the singularities is obtained. Furthermore, in accordance with the centre manifold theorem, the subcritical Hopf bifurcation is analysed and obtained. Numerical simulations demonstrate the correctness of the dynamical and bifurcation analyses. Moreover, by choosing appropriate parameters, this perturbed system can exhibit chaotic, quasiperiodic and periodic dynamics, as well as some coexisting attractors, such as a chaotic attractor coexisting with a periodic attractor for \(a>0\), and a chaotic attractor coexisting with a quasiperiodic attractor for \(a=0\). Coexisting attractors are not associated with an unstable equilibrium and thus often go undiscovered because they may occur in a small region of parameter space, with a small basin of attraction in the space of initial conditions.     

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.     

Period-adding bifurcations and chaos in a bubble column

We obtained period-adding bifurcations in a bubble formation experiment. Using the air flow rate as the control parameter in this experiment, the bubble emission from the nozzle in a viscous fluid undergoes from single bubbling to a sequence of periodic bifurcations of k to k+1 periods, occasionally interspersed with some chaotic regions. Our main assumption is that this period-adding bifurcation in bubble formation depends on flow rate variations in the chamber under the nozzle. This assumption was experimentally tested by placing a tube between the air reservoir and the chamber under the nozzle in the bubble column experiment. By increasing the tube length, more period-adding bifurcations were observed. We associated two main types of bubble growth to the flow rate fluctuations inside the chamber for different bubbling regimes. We also studied the properties of piecewise nonlinear maps obtained from the experimental reconstructed attractors, and we concluded that this experiment is a spatially extended system.     

The dynamics of a symmetric coupling of three modified quadratic maps

We investigate the dynamical behavior of a symmetric linear coupling of three quadratic maps with exponential terms, and identify various interesting features as a function of two control parameters. In particular, we investigate the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable period-l, period-2, and period-3 orbits. We also investigate the multistability in the same coupling. Lyapunov exponents, parameter planes, phase space portraits, and bifurcation diagrams are used to investigate transitions from periodic to quasiperiodic states, from quasiperiodic to mode-locked states and to chaotic states, and from chaotic to hyperchaotic states.