The dynamical origin of the universality classes of spatiotemporal intermittency

Two universality classes of spatiotemporal intermittency are seen in the spreading and non-spreading regimes of the sine circle map lattice, spatiotemporal intermittency of the directed percolation class, and spatial intermittency, not of the DP class, where the temporal behavior is regular. The transition between the two classes maps to a probabilistic to deterministic transition of the equivalent cellular automaton of the model, and is seen to have its dynamic origin in an attractor-widening crisis.     

Probabilistic signatures of spatiotemporal intermittency in the coupled sine circle map lattice

The phase diagram of the coupled sine circle map system exhibits a variety of interesting phenomena including spreading regions with spatiotemporal intermittency, non-spreading regions with spatial intermittency, and coherent structures termed solitons. A spreading to non-spreading transition is seen in the system. A cellular automaton version of the coupled system maps the spreading to non-spreading transition to a transition from a probabilistic to a deterministic cellular automaton. The solitonic sector of the system shows spatiotemporal intermittency with soliton creation, propagation and absorption. A probabilistic cellular automaton mapping is set up for this sector which can identify each one of these phenomena.      

Chimera states in globally coupled sine circle map lattices: Spatiotemporal intermittency and hyperchaos

Chimera states with coexisting phase synchronised and desynchronised groups are seen in a globally coupled sine circle map lattice with distinct values of intergroup and intragroup coupling, at some parameters. Structures similar to spatiotemporally intermittent regions, are seen in the desynchronised group. These states exhibit hyperchaos. Other states seen include fully synchronised/desynchronised states and two cluster states. The phase diagram of the system is obtained using complex order parameters. This is reproduced accurately using the average fraction of laminar/burst sites in each group, indicating the key nature of this quantity.     

Transitional Channel Flow: A Minimal Stochastic Model

In line with Pomeau’s conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes observed in channel flow before its return to laminar flow. Numerical simulations show that a regime with bands obliquely drifting in two stream-wise symmetrical directions bifurcates into an asymmetrical regime, before ultimately decaying to laminar flow. The model is expressed in terms of a probabilistic cellular automaton of evolving von Neumann neighborhoods with probabilities educed from a close examination of simulation results. It implements band propagation and the two main local processes: longitudinal splitting involving bands with the same orientation, and transversal splitting giving birth to a daughter band with an orientation opposite to that of its mother. The ultimate decay stage observed to display one-dimensional DP properties in a two-dimensional geometry is interpreted as resulting from the irrelevance of lateral spreading in the single-orientation regime. The model also reproduces the bifurcation restoring the symmetry upon variation of the probability attached to transversal splitting, which opens the way to a study of the critical properties of that bifurcation, in analogy with thermodynamic phase transitions.     

Directed percolation phase transition at the onset of STI in an inhomogeneous coupled map lattice

A model for inhomogeneously coupled logistic maps is considered to find some critical exponents in the transition from inhomogeneous steady state to spatiotemporal chaos through spatiotemporal intermittency. The laminar state in the model is described by inhomogeneous steady state with spatial period two. We obtain a complete set of static exponents which match with the corresponding directed percolation (DP) values in (1+1) dimension. We also find four nonuniversal spreading exponents in which three exponents are in agreement with DP values. The model in which absorbing state is inhomogeneous steady state, contributes a new example in evidence of Pomeau's [18] conjecture that the onset of STI in a deterministic system belongs to DP universality class.     

Observations and modeling of deterministic properties of human heart rate variability

Simple models show that in Type-I intermittency a characteristic U-shaped probability distribution is obtained for the laminar phase length. The laminar phase length distribution characteristic for Type-I intermittency may be obtained in human heart rate variability data for some cases of pathology. The heart and its regulatory systems are presumed to be both noisy and non-stationary. Although the effect of additive noise on the laminar phase distribution in Type-I intermittency is well-known, the effect of neither multiplicative noise nor non-stationarity have been studied. We first discuss the properties of two classes of models of Type-I intermittency: (a) the control parameter of the logistic map is changed dichotomously from a value within the intermittency range to just below the bifurcation point and back; (b) the control parameter is changed randomly within the same parameter range as in the model class (a). We show that the properties of both models are different from those obtained for Type-I intermittency in the presence of additive noise. The two models help to explain some of the features seen in the intermittency in human heart rate variability.     

Circuit implementation and multiform intermittency in ahyper-chaotic model extended from Lorenz system

<正>This paper presents a non-autonomous hyper-chaotic system,which is formed by adding a periodic driving signal to a four-dimensional chaotic model extended from the Lorenz system.The resulting non-autonomous hyper-chaotic system can display any dynamic behaviour among the periodic orbits,intermittency,chaos and hyper-chaos by controlling the frequency of the periodic signal.The phenomenon has been well demonstrated by numerical simulations,bifurcation analysis and electronic circuit realization.Moreover,the system is concrete evidence for the presence of Pomeau-Manneville Type-Ⅰintermittency and crisis-induced intermittency.The emergence of a different type of intermittency is similarly subjected to the frequency of periodic forcing.By statistical analysis,power scaling laws consisting in different intermittency are obtained for the lifetime in the laminar state between burst states.     

Breakdown of universality in transitions to spatiotemporal chaos.

We show that the transition from laminar to active behavior in extended chaotic systems can vary from a continuous transition in the universality class of directed percolation with infinitely many absorbing states to what appears as a first-order transition. The latter occurs when finite lifetime nonchaotic structures, called "solitons," dominate the dynamics. We illustrate this scenario in an extension of the deterministic Chaté-Manneville coupled map lattice model and in a soliton including variant of the stochastic Domany-Kinzel cellular automaton.     

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations     

Coupled map lattices as cellular automata

We approach the problem of the complex dynamics of coupled map lattices (CML) by proposing a reduction to deterministic cellular automata (CA) with more than two states per site. The reduction scheme replaces the local map by an approximation in terms of a step function based on a straightforward analysis of the local dynamics. The variation of the spatial coupling in the CML then translates itself as a path in the spaces of rules for the equivalent deterministic CA. The transition to turbulence via spatiotemporal intermittency in the CML is then interpreted as a transition in the space of rules. The observed nonuniversality of this transition can be traced back to the nature of the rules involved on both sides of the transition region and to the character of the escape process from the turbulent state, either strongly deterministic or quasiprobabilistic. The relation between CML, deterministic, and probabilistic CA and the possibility of a mean-field treatment of the dynamics of CML are discussed at a more formal level.     

Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example.     

Hénon映射的噪声诱导间歇现象及其临界值估算研究

本文研究了Hénon映射在噪声诱导下发生的间歇现象.通过数值模拟和全局分析手段,揭示了噪声诱导间歇现象的机理.基于随机敏感度函数法,通过检测噪声作用下周期吸引子的置信椭圆与混沌鞍的碰撞情况,给出了诱发间歇现象的噪声强度临界值的估算方法.结果表明,Hénon映射中噪声诱导间歇现象是由随机周期吸引子和混沌鞍不稳定流形的相互作用引发,随机敏感度函数的方法可以较好地估算发生间歇现象的噪声强度临界值.     

Snaking and isolas of localised states in bistable discrete lattices

We consider localised states in a discrete bistable Allen-Cahn equation. This model equation combines bistability and local cell-to-cell coupling in the simplest possible way. The existence of stable localised states is made possible by pinning to the underlying lattice; they do not exist in the equivalent continuum equation. In particular we address the existence of ‘isolas’: closed curves of solutions in the bifurcation diagram. Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions. We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D.     

Mode-locking of self-generated oscillations in a semiconductor model for low-temperature impurity breakdown

The nonlinear dynamic behaviour of a model for self-generated, spatially coupled current oscillations in two separated parts of a semiconductor is analysed. The model involves impurity impact ionization, nonlinear energy relaxation of hot carriers, and energy exchange between the two subsystems. Quasiperiodicity and mode-locking are obtained, and characterized by a suitably defined rotation number and a spectral bifurcation diagram. The mode-locking structure is found to obey the Farey tree ordering, and can be understood on the basis of the circle map theory, assuming a particular path in the two-dimensional phase diagram of the circle map.     

The emergence of coherent structures in coupled map lattices

We show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices. This phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics. If we further increase the coupling then, new ground states appear showing the coexistence of a large-scale coherent structure with a small-scale chaotic motion. This allows us to propose a generalization of the notion of spatial intermittency.     

Intermittency of the density fluctuations and its influence on the radial transport in the boundary of J-TEXT

To improve the understanding of the turbulence intermittency,a detailed investigation of the intermittency of the density fluctuations has been performed in the boundary of J-TEXT.The intermittency of the density fluctuations and its influence on the radial transport are reported.The probability distribution functions of the density fluctuations are not scale-invariant,being inconsistent with the self-organized criticality hypothesis.The underlying dynamics of the intermittency are detected using the quiet-time statistical method.The probability distribution function of the quiet times shows double-power-law regions,indicating the existence of correlations between the successive burst events.     

Modification of instability processes by multiplicative noises

We study two dynamical systems submitted to white and Gaussian random noise acting multiplicatively. The first system is an imperfect pitchfork bifurcation with a noisy departure from onset. The second system is a pitchfork bifurcation in which the noise acts multiplicatively on the non-linear term of lowest order. In both cases noise suppresses some solutions that exist in the deterministic regime. Besides, for the first system, the imperfectness of the bifurcation reduces the regime of on-off intermittency. For the second system, the unstable mode can achieve a jump of finite amplitude at instability but without hysteresis. We finally identify a generic property that is verified by the stationary probability density function of the dynamical variable when a control parameter is varied.     

Statistical analysis of the transition to turbulence in plane Couette flow

We argue on general grounds that the transition to turbulence in plane Couette flow is best studied experimentally at a statistical level. We present such a statistical analysis of experimental data guided by a parallel investigation of a simple coupled map lattice model for spatiotemporal intermittency. We confirm that this generic type of spatiotemporal chaos is relevant in the context of plane Couette flow, where the linear stability of the laminar regime at all Reynolds numbers insures the necessary local subcriticality. Using large ensembles of similar experiments, we show the existence of a well-defined threshold Reynolds number above which a unique, turbulent, intermittent attractor coexists with the laminar flow. Furthermore, our data reveals that this transition to spatiotemporal intermittency is discontinuous, i.e. akin to a first-order phase transition. Received: 10 April 1998 / Revised: 22 June 1998 / Accepted: 24 June 1998     

Traffic dynamics of an on-ramp system with a cellular automaton model

This paper uses the cellular automaton model to study the dynamics of traffic flow around an on-ramp with an acceleration lane. It adopts a parameter, which can reflect different lane-changing behaviour, to represent the diversity of driving behaviour. The refined cellular automaton model is used to describe the lower acceleration rate of a vehicle. The phase diagram and the capacity of the on-ramp system are investigated. The simulation results show that in the single cell model, the capacity of the on-ramp system will stay at the highest flow of a one lane system when the driver is moderate and careful; it will be reduced when the driver is aggressive. In the refined cellular automaton model, the capacity is always reduced even when the driver is careful. It proposes that the capacity drop of the on-ramp system is caused by aggressive lane-changing behaviour and lower acceleration rate.     

含指数项广义平方映射的分岔和吸引子

由平方映射延伸构造出了一类含指数项的广义平方映射,并由一维映射通过一次耦合项得到了二维映射.利用一参数分岔图、二参数动力学行为分布图、映射迭代曲线和吸引子相图等方法对这类广义平方映射进行了分析和仿真.研究结果表明：一维广义平方映射分布在一个单位区域内的,有着与单峰平方映射相类似的非线性动力学现象;而二维广义平方映射则存在Hopf分岔和锁频等现象,有着复杂多变、形状奇异的极限环和混沌吸引子. 关键词： 广义平方映射 分岔 迭代曲线 吸引子