Session Y53: Nonlinear Dynamics of Coupled Systems

Frequency Enhancement in Coupled Noisy Excitable Elements

The oscillatory dynamics of coupled noisy excitable FitzHugh-Nagumo elements is investigated as a function of the coupling strength~$g$. For two such coupled elements, it~is found that their frequencies are enhanced by the coupling and will synchronize at a frequency~higher than the uncoupled frequencies of each element. As~$g$ increases, there is an unexpected peak~in the mean of frequency distribution~before reaching synchronization at the optimal coupling strength.~This phenomenon~can be understood with a simplified analytic model~based on the excitation~across a potential barrier whose height is controlled by $g$~as well as the formation of temporary coherent cluster.   

Ramification of stream networks

The geometric complexity of channel networks arises from their successive ramifications --- the splitting of a single tip into two branches. Here we show that streams incised by groundwater seepage split at a characteristic angle of $\alpha=2\pi/5=72^\circ$. Our theory represents streams as a collection of paths growing and bifurcating in a diffusing field, which can be described by Loewner dynamics. Analysis of thousands of bifurcations in a $\sim$100~km$^2$ stream network reveals that the mean branching angle is $72.5^\circ\pm 1.5^\circ$ (95\% C.I.) and that the five fold symmetry induced by the branching of the tips is observed on all scales in the network. This consistency between theory and observation suggests that the network geometry is determined by the external flow field rather than flow within the streams themselves, contrary to assumptions made by models that relate geometry to internal dissipation.   

Dynamical phyllotaxis: transition modes and Sine-Gorden-like solitons

Repulsive particles constrained to a cylindrical surface generate a rich set of static spiral patterns known as phyllotaxis. The ground states and quasi-static transitions of phyllotaxis has been well understood based on the conventional hypothesis that particles always form a cylindrical lattice. We investigate inhomogeneous transition modes which break this helical symmetry to connect monojugate and multijugate patterns. Furthermore, traveling Sine-Gordon-like solitons in dynamical phyllotaxis are observed in numerical simulations and explained using a modified Frenkel-Kontorova model. We show that kinks propagate spirally along selected parastichies and carry an intrinsic dipole. Applications in different areas of physics are discussed.   

Cascade of bifurcation type trajectories as a general type of attractors in fractional dynamical systems

Based on the results of computer simulations and analytical investigation of fractional maps, we present our latest results related to the existence and stability of the new type of attractors in the fractional dynamical systems: cascade of bifurcation type trajectories (CBTT). We show that in fractional Standard Map (FSM) this type of attractors appears in the area of parameters, where in the corresponding integer system (regular Standard Map) series of period doubling bifurcations and corresponding splitting of islands of stability leads to the disappearance of stability and transition to chaos.   

Scale-free avalanches in disordered systems of localized charges with long-range Coulomb interaction

We study theoretically and numerically the charge avalanches created by a perturbation in disordered systems of localized charges with unscreened Coulomb interaction (the so-called electron glass model), in two and three dimensions. Starting from a low-lying local energy minimum, we perturb the system by inserting an extra charge or an extra dipole, and let it relax via one-particle hops until a new minimum is reached. We find that the size distribution of the avalanches created in this process displays generically a power-law tail with an exponent close to the mean-field value 3/2 both in 2D and 3D, without requiring any parameter tuning. We provide a qualitative explanation of these results in terms of the density of states of elementary charge and dipole excitations and the associated Coulomb gap, which shows that the power-law tail arises from arbitrarily long hops, without requiring to assume the existence of a glass phase. Finally, we discuss the experimental relevance of these results and compare our picture to similar scale-free avalanches observed in mean field spin glasses, in which they are are associated to a marginal glass phase.   

Cooperation-induced temporal complexity in networks of pulse-coupled units

We study a network of stochastic pulse-coupled units generating bursts with the same size distribution as the neuronal avalanches in mature cultured neurons, recently revealed by the experimental observation. We prove that in addition to this form of complexity this model yields a form of phase transition generating also temporal complexity. This means that the distance from two consecutive bursts fits the prescription of a Mittag-Leffler (ML) function renewal theory. There exists a critical value of the cooperation parameter at which this description applies to the whole time regime. By increasing the cooperation parameter the ML theory breaks down and the sequence of bursts tend to become periodic with the same intensity. We make the conjecture that the analysis of this model may shed light into the theoretical foundation of neuronal burst leaders and that the recently discovered principle of complexity management may be conveniently applied to the neuro-physiological processes that are properly described by this model.   

Time dependence of reprecipitation rates in heterogeneous media

The analysis of spatial and temporal variations of the chemical and isotopic compositions of minerals in sedimentary systems provides a powerful tool for calculating dissolution and reprecipitation rates, and has previously been applied to find time-dependent rates in marine sediments. Dissolution and precipitation processes tend to shift the composition of the pore fluids toward that of the solid phase, and vice-versa. Current theory treats both the fluid and solid phases as well-mixed reservoirs, relying on mean-field theory that is inconsistent with the physical structure of the solid, as dissolution and precipitation occur only on the reactive surface of the solid. We present a model that accounts for the heterogeneity of the solid phase by adding and removing material only at the reactive surface. We therefore model the location of the surface with a 1-D random walk, in which the buried bulk of the solid phase can only be modified through repeated dissolution events. We approximate this physical scenario with a three-reservoir kinetic model and more detailed numerical simulations. We develop an understanding of two power-law scaling regimes, the second of which demonstrates 1/time aging in the rate constant, similar to those observed in marine sediment studies.   

Properties of compacton-anticompacton collisions

We study the properties of compacton-anticompacton collision processes. We compare and contrast results for the case of compacton-anticompacton solutions of the K(l,p) Rosenau-Hyman (RH) equation for l=p=2, with compacton-anticompacton solutions of the L(l,p) Cooper-Shepard-Sodano (CSS) equation for p=1 and l=3. This study is performed using a Pad\'e discretization of the RH and CSS equations. We find a significant difference in the behavior of compacton-anticompacton scattering. For the CSS equation, the scattering can be interpreted as ``annihilation'' as the wake left behind dissolves over time. In the RH equation, the numerical evidence is that multiple shocks form after the collision, which eventually lead to ``blowup'' of the resulting wave form.   

Cardiac arrhythmias and degradation into chaotic behavior prevention using feedback control

During normal heart rhythm, cardiac cells behave as a set of oscillators with a distribution of phases but with the same frequency. The heart as a dynamical system in a phase space representation can be modeled as a set of oscillators that have closed overlapping orbits with the same period. These orbits are not stable and in the case of disruption of the cardiac rhythm, such as due to premature beats, the system will have a tendency to leave its periodic unstable orbits. If these orbits become attracted to phase singularities, their disruption may lead to chaotic behavior, which appears as a life-threating ventricular fibrillation. By using closed-loop feedback in the form of an adjustable defibrillation shock, any drift from orbits corresponding to the normal rhythm can be corrected by forcing the system to maintain its orbits. The delay through the feedback network coincides with the period of normal heart beats. To implement this approach we developed a 1 kW arbitrary waveform voltage-to-current converter with a 1 kHz bandwidth driven by a photodiode system that records an optical electrocardiogram and provides a feedback signal in real time. Our goal is to determine whether our novel method to defibrillate the heart will require much lower energies than are currently utilized in single shock defibrillators.   

Experiments on oscillator ensemble with global nonlinear coupling

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a linear or nonlinear phase-shifting unit in the global feedback loop. With linear unit we observe, with increase of the coupling strength, a standard Kuramoto-like transition to a fully synchronous state; the threshold of the transition depends on the phase shift. In case of nonlinear global coupling we first observe a transition to a state when approximately half of the population forms a synchronous cluster. With further increase of the coupling strength we observe destruction of this cluster and formation of a self-organized quasiperiodic state, predicted in [M. Rosenblum and A. Pikovsky, PRL, 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. The transition is characterized by a non-monotonic dependence of the order parameter on the coupling strength. We demonstrate a good correspondence between theory and experiment.   

Dynamics of Confident Voting

In the classical voter model, a voter has no intrinsic confidence in its current opinion. We introduce the confident voter model in which each voter can be in one of two opinions, and can additionally have two levels of commitment to an opinion --- confident and vacillating. Upon interacting with an agent of a different opinion, a confident voter becomes less committed, or vacillating, but does not immediately change opinion. However, a vacillating agent changes opinion upon interacting with an agent of a different opinion. In the mean-field limit, a population of size $N$ is quickly driven to a mixed state before consensus is eventually achieved in a time of order $\ln{N}$. In two dimensions, the distribution of consensus times is characterized by two distinct times --- one that scales linearly with $N$ and another that scales as $N^{3/2}$. The longer time arises from configurations that fall into long-lived stripe states, which are caused by an effective surface tension between domains of different opinion states, before consensus is finally reached.   

Nonuniversal Effects in Mixing Correlation-Growth Processes with Randomness

In mixed-growth dynamical process $P = Y \vee X$ there are two dynamical processes: $Y$ (in one universality class) and $X$ (in another universality class). They \textit{alternate} with each other: ``\textit{exclusively either} $Y$ (is active with probability $q$) \textit{or} $X$ (is active with probability $p$),'' $p+q=1$. When $P$ models surface growth via deposition/desorption/adsorption with $X$ building universal correlations and $Y$ representing randomness (e. g., thermal effects), in order to correctly construct a continuum growth equation for $P$ a distinction must be made \textit{within a single universality class} of $X$ between processes that do and do not create voids in the bulk of deposited material. Then, model-dependent prefactors in universal scaling of the surface roughness can be linked with the bulk morphology and determined from the bulk structures. This connection is essential to finding correct dynamical scaling and to interpretation of scaling laws for mixed-growth dynamical processes.   

Non-equilibrium modulated phases in a system with local energy input

The equilibrium phase diagram of the two-dimensional Ising model in contact with a single heat bath is well understood. We here study the properties of the two-dimensional Ising model with conserved dynamics where the two halves of the system are in contact with different heat baths. Using Monte Carlo simulations, we identify three different phases for this non-equilibrium system, as a function of the aspect ratio of the lattice and of the temperatures. The first phase is characterized by the complete disorder of the particles, while the second phase is characterized by the complete order of the particles. The third phase is the most interesting one as it displays stripes with widths that depend on the system parameters. The full phase diagram of our non-equilibrium system is determined through the study of the structure factor.   

Nonequilibrium relaxation and critical aging for driven Ising lattice gases

We employ Monte Carlo simulations to study the non-equilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density auto-correlation function in the non-equilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time auto-correlations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice gases, valid to all orders in the dimensional expansion.