Institute of Mathematics

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatiotemporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rössler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence.

The discrete infinite-dimensional modal system describing nonlinear sloshing of an incompressible fluid with irrotational flow partially occupying a tank performing an arbitrary three-dimensional motion is derived in general form. The tank has vertical walls near the free surface and overturning waves are excluded. The derivation is based on the Bateman–Luke variational principle. The free surface motion and velocity potential are expanded in generalized Fourier series. The derived infinite-dimensional modal system couples generalized time-dependent coordinates of free surface elevation and the velocity potential. The procedure is not restricted by any order of smallness. The general multidimensional structure of the equations is approximated to analyse sloshing in a rectangular tank with finite water depth. The amplitude–frequency response is consistent with the fifth-order steady-state solutions by Waterhouse (1994). The theory is validated by new experimental results. It is shown that transients and associated nonlinear beating are important. An initial variation of excitation periods is more important than initial conditions. The theory is invalid when either the water depth is small or water impacts heavily on the tank ceiling. Alternative expressions for hydrodynamic loads are presented. The procedure facilitates simulations of a coupled vehicle–fluid system.

Abstract Pollen and plant macrofossil data from northern Eurasia were used to reconstruct the vegetation of the last glacial maximum (LGM: 18,000 ± 2000 14 C yr bp ) using an objective quantitative method for interpreting pollen data in terms of the biomes they represent ( Prentice et al. , 1996 ). The results confirm previous qualitative vegetation reconstructions at the LGM but provide a more comprehensive analysis of the data. Tundra dominated a large area of northern Eurasia (north of 57°N) to the west, south and east of the Scandinavian ice sheet at the LGM. Steppe‐like vegetation was reconstructed in the latitudinal band from western Ukraine, where temperate deciduous forests grow today, to western Siberia, where taiga and cold deciduous forests grow today. The reconstruction shows that steppe graded into tundra in Siberia, which is not the case today. Taiga grew on the northern coast of the Sea of Azov, about 1500 km south of its present limit in European Russia. In contrast, taiga was reconstructed only slightly south of its southern limit today in south‐western Siberia. Broadleaved trees were confined to small refuges, e.g. on the eastern coast of the Black Sea, where cool mixed forest was reconstructed from the LGM data. Cool conifer forests in western Georgia were reconstructed as growing more than 1000 m lower than they grow today. The few scattered sites with LGM data from the Tien‐Shan Mountains and from northern Mongolia yielded biome reconstructions of steppe and taiga, which are the biomes growing there today.

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

The authors study the best possible accuracy of recovering the solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the solution is expressed in terms of general source conditions, given through index functions. The emphasis is on geometric concepts.

Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states," has been described, where in a spatially homogeneous system, regions of irregular incoherent motion coexist with regular synchronized motion, forming a self-organized pattern in a population of nonlocally coupled oscillators. Whereas most previous studies of chimera states focused their attention on the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, here we investigate the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimera's cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newton's laws that are ubiquitous in many physical and engineering systems.

Bau-Sen Du introduced a notion of chaos which is stronger than Li-Yorke sensitivity. A TDS (X, f ) is called chaotic if there is a positive such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not -asymptotic. In this article, we show that a TDS (T, f ) is transitive but not mixing if and only if (T, f ) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.

Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Infinite tank roof height and no overturning waves are assumed. The modal theory is based on an infinite-dimensional system of nonlinear ordinary differential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an infinite-dimensional system of ordinary differential equations with fifth-order polynomial nonlinearity by assuming sufficiently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in different domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l . Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h / l = 0.173, where the previous model by Faltinsen et al . (2000) failed. The new model agrees well with experiments.

The theory of parabolic equations, a well-developed part of the contemporary partial differential equations and mathematical physics, is the subject theory of of an immense research activity. A contin

We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interactions, where each element is coupled to its nearest neighbors within a finite range. Depending upon the coupling strength and coupling radius, we find characteristic spatial patterns such as wavelike profiles and study the transition from coherence to incoherence leading to spatial chaos. We analyze the origin of this transition based on numerical simulations and support the results by theoretical derivations, identifying a critical coupling strength and a scaling relation of the coherent profiles. To demonstrate the universality of our findings, we consider time-discrete as well as time-continuous chaotic models realized as a logistic map and a Rössler or Lorenz system, respectively. Thereby, we establish the coherence-incoherence transition in networks of coupled identical oscillators.

An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. (2000) for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) ‘planar’/‘diagonal’ resonant standing waves for longitudinal/diagonal forcing, (ii) ‘swirling’ waves moving along tank walls clockwise or counterclockwise and (iii) ‘square’-like resonant standing wave coupling in-phase oscillations of both the lowest modes. The frequency domains for stable and unstable waves (i)–(iii), the contribution of higher modes and the influence of decreasing fluid depth are studied in detail. The zones where either unstable steady regimes exist or there are two or more stable periodic solutions with similar amplitudes are found. New experimental results are presented and show generally good agreement with theoretical data on effective domains of steady-state sloshing. Three-dimensional sloshing regimes demonstrate a significant contribution of higher modes in steady-state and transient flows.

Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

In this tutorial paper, various phenomena linked to the synchronization of chaotic systems are discussed using the simple example of two coupled skew tent maps. The phenomenon of locally riddled basins of attraction is explained using the Lyapunov exponents transversal to the synchronization manifold. The skew tent maps are coupled in two different ways, leading to quite different global dynamic behavior especially when the ideal system is perturbed by parameter mismatch or noise. The linear coupling leads to intermittent desynchronization bursts of large amplitude, whereas for the nonlinear coupling the synchronization error is asymptotically uniformly bounded.

We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six, and ten indistinguishable oscillators, where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks.

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h / l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h / l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tank's surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h / l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Chimera states are a recently new discovered dynamical phenomenon that appears in arrays of nonlocally coupled oscillators and displays a spatial pattern of coherent and incoherent regions. We report here an additional feature of this dynamical regime: an irregular motion of the position of the coherent and incoherent regions, i.e., we reveal the nature of the chimera as a spatiotemporal pattern with a regular macroscopic pattern in space, and an irregular motion in time. This motion is a finite-size effect that is not observed in the thermodynamic limit. We show that on a large time scale, it can be described as a Brownian motion. We provide a detailed study of its dependence on the number of oscillators N and the parameters of the system.

We present a simplified phase model for neuronal dynamics with spike timing-dependent plasticity (STDP). For asymmetric, experimentally observed STDP we find multistability: a coexistence of a fully synchronized, a fully desynchronized, and a variety of cluster states in a wide enough range of the parameter space. We show that multistability can occur only for asymmetric STDP, and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies. We test the efficacy of the proposed method on the Kuramoto model which is, de facto, one of the sample models for a description of the phase dynamics in neuronal ensembles.

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherence-incoherence patterns and to suggest directions for searching new ones.