Session S22: Nonlinear Dynamics and Applications

Stochastic ``Time''

We present a simple dynamical model which uses ``non-locality'' and ``noise'' on time axis. The model is a delayed dynamical map model with a stochasticity on the variable corresponding to ``time'' steps. The analogy is made with a tape recorder whose recording devise can move back on a tape as it records the values of the dynamical variable. With a tuned probability of ``moving backward'' with a given delay, the dynamics of the model shows an oscillatory behavior, similar to the one found in the models of stochastic resonance. We discuss implication of this model and whether it provides any reasonable approach to considering ``non-locality'' and ``noise'' on time axis. \newline \newline References: \newline [1] T. Ohira, ArXive: cond-mat/0607544 (To appear in the AIP conf. proc. of the 8th Int. Symp. on Frontiers of Fundamental Physics, Madrid, Spain, October 17-19 2006.) \newline [2] T. Ohira, ArXive: cond-mat/0610032 (To appear in the AIP conf. proc. of the 9th Granada Seminar, Granada, Spain, September 11-15, 2006.) \newline [3] T. Ohira and Y. Sato, PRL 82, 2811 (1999).   

Towards a RMT Scattering-matrix with universal frequency correlations

We concern ourselves with the prediction of mesoscopic wave phenomena from statistical knowledge of classical trajectories. A diffusing particle picture for the flow of mean probability in chaotic systems is used to estimate dynamical features of mean square time-domain S-matrices for waves coupled in and out through one perfectly open channel. The additional constraint of unitarity and minimum phase, then leads to a unique and plausible S-matrix that exhibits familiar mesoscopic wave dynamics. These include enhanced backscatter, quantum echo, power law tails, level repulsion and spectral rigidity. We conjecture that a generalization to n x n S matrices would exhibit behavior identical to that of the GOE or GUE depending on its symmetries.   

Synthesizing Chaos

Chaos is usually attributed only to nonlinear systems. Yet it was recently shown that chaotic waveforms can be synthesized by linear superposition of randomly polarized basis functions. The basis function contains a growing oscillation that terminates in a large pulse. We show that this function is easily realized when viewed backward in time as a pulse followed by ringing decay. Consequently, a linear filter driven by random pulses outputs a waveform that, when viewed backward in time, exhibits essential qualities of chaos, i.e. determinism and a positive Lyapunov exponent. This phenomenon suggests that chaos may be connected to physical theories whose framework is not that of a deterministic dynamical system. We demonstrate that synthesizing chaos requires a balance between the topological entropy of the random source and the dissipation in the filter. Surprisingly, using different encodings of the random source, the same filter can produce both Lorenz-like and R\"{o}ssler-like waveforms. The different encodings can be viewed as grammar restrictions on a more general encoding that produces a chaotic superset encompassing the Lorenz and R\"{o}ssler paradigms of nonlinear dynamics. Thus, the language of deterministic chaos provides a useful description for a class of signals not generated by a deterministic system.   

Eigenvalues of the time evolution operator governing nuclear spin behavior in solids

The decay of nuclear magnetic resonance (NMR) signals in solids is an extremely difficult many-body problem with no complete solution. Utilizing frozen xenon polarized by spin-exchange optical pumping, we have observed the long-time behavior of the transverse NMR signal for both free-induction decay and spin (solid) echoes. The hyperpolarized signal can be observed for up to $\sim $10 decay constants, allowing us to characterize the long-time behavior, which is predicted to have one of two forms: $S(t)\sim e^{-\gamma t}$ or $S(t)\sim e^{-\gamma t}\cos (\omega t+\phi )$, where the constants $\omega $ and $\gamma $ are the same for the FID as for the solid echo. Our data agree well with this prediction, which follows from considering the evolution of the density matrix under the action of its time evolution operator, with the corresponding eigenvalues determining the evolution of the spin system.* Not only is this decay an example of Markovian behavior on non-Markovian timescales but these eigenvalues should be a deep fundamental property of many-body quantum systems. The eigenvalues are also expected to be analogous to Pollicott-Ruelle resonances in classical chaotic systems. *B.V. Fine, Phys. Rev. Lett. 94, 247601 (2005) .   

Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos

Symbolic dynamics has proven to be an invaluable tool in analyzing the mechanisms that lead to unpredictability and random behavior in nonlinear dynamical systems. Surprisingly, a discrete partition of continuous state space can produce a coarse-grained description of the behavior that accurately describes the invariant properties of an underlying chaotic attractor. In particular, measures of the rate of information production--the topological and metric entropy rates--can be estimated from the outputs of Markov or generating partitions. Here we develop Bayesian inference for k-th order Markov chains as a method for finding generating partitions and estimating entropy rates from finite samples of discretized data produced by coarse-grained dynamical systems.   

Nonlinear Dynamics of Nanomechanical Resonators

Nanoelectromechanical systems (NEMS) offer great promise for many applications including motion and mass sensing. Recent experimental results suggest the importance of nonlinear effects in NEMS, an issue which has not been addressed fully in theory. We report on a nonlinear extension of a recent analytical model by Armour et al [1] for the dynamics of a single-electron transistor (SET) coupled to a nanomechanical resonator. We consider the nonlinear resonator motion in both (a) the Duffing and (b) nonlinear pendulum regimes. The corresponding master equations are derived and solved numerically and we consider moment approximations as well. In the Duffing case with hardening stiffness, we observe that the resonator is damped by the SET at a significantly higher rate. In the cases of softening stiffness and the pendulum, there exist regimes where the SET adds energy to the resonator. To our knowledge, this is the first instance of a single model displaying both negative and positive resonator damping in different dynamical regimes. The implications of the results for SET sensitivity as well as for, as yet unexplained, experimental results will be discussed. 1. Armour et al. Phys.Rev.B (69) 125313 (2004).   

Interference of fractals - a method to control the deterministic stochastic multiresonance

We present a new method to control the deterministic stochastic multiresonance in dynamical systems, which can be considered as a threshold-crossing systems, in the vicinity of chaotic crises. As an example we choose a two-dimensional chaotic map, where the threshold-crossing probability follows the overlap of the fractal structures of chaotic saddles and the basins of escape. Using a small periodic perturbation we induce interference like behaviour in fractal structure leading to significant changes of the information transmission through the system. The analytical theory based on topological model is in a reasonable agreement with the numerical results for mutual information between the input and output signal.   

Arnold Tongue Mixed Reality States in an Interreality System

We present experimental data on the limiting behavior of an interreality system comprising a virtual horizontally driven pendulum coupled to its real-world counterpart, where the interaction time scale is much shorter than the time scale of the dynamical system. We present experimental evidence that if the physical parameters of the virtual system match those of the real system within a certain tolerance, there is a transition from an uncorrelated dual reality state to a mixed reality state of the system in which the motion of the two pendula is highly correlated. The region in parameter space for stable solutions has an Arnold tongue structure for both the experimental data and for a numerical simulation. As virtual systems better approximate real ones, even weak coupling in other interreality systems may produce sudden changes to mixed reality states. This work was supported by the National Science Foundation Grant No. NSF PHY 01-40179, NSF DMS 03-25939 ITR, and NSF DGE 03-38215.   

Complex Dynamics in Systems of Interacting Bosons

We consider interacting bosons described by a Bose-Hubbard Hamiltonian (BHH) and analyze the evolving energy distribution as an experimentally controllable parameter, the coupling strength k between neighboring sites, is changed. Three driving schemes of k are considered: (a) the sudden limit (LDoS analysis), (b) the one-pulse scheme (wavepacket dynamics), and (c) the time-reversal scheme (fidelity). We find in all cases two distinct regimes: the Linear Response (LRT) regime where we can trust the Fermi-Golden-Rule picture, and what we call the non-perturbative regime where the perturbation k is quantum mechanically large. In the former regime, the evolving distribution can be described by an improved Random Matrix Theory (RMT) which takes into accont the structured energy landscape of the perturbation. Instead, in the latter regime, non-universal features of the underlying classical dynamics dictate the energy spreading thus leading to a clash with the predictions of RMT.   

Synchronization in a chain of nearest neighbors coupled oscillators with fixed ends

A system of coupled phase oscillators with nearest neighbors coupling in a chain with fixed ends is investigated. The system synchronizes to a common value of the time averaged frequency which depends on the initial phases of the oscillators at the ends of the chain. The time-averaged frequency decays as the coupling strength increases .Near the transition to the frozen state, the time-averaged frequency has a power law behavior as a function of the coupling strength with synchronized time averaged frequency equal to zero. Associated with this power law, there is an increase in phases of each oscillator with specific jumps with a scaling law of the elapsed time between the jumps. During the interval between the full frequency synchronization and the transition to the frozen state, the maximum Lyapunov exponent indicates quasiperiodicity. Time series analysis of the oscillators frequency shows this quasiperiodicity as the coupling strength increases.   

Stationary and traveling solitons in one-dimensional quartic lattices

We discuss the solutions to classical vibrations of a monatomic one-dimensional lattice. The interaction potential between the nearest neighbor atoms in the lattice contains nonlinear quartic terms. We found a total of $N$ normal modes, that are symmetric and antisymmetric with respect to the center of the chain consisting of $N$ atoms. Also, there exist stationary soliton solutions that are neither symmetric nor antisymmetric suggesting the total number of solutions exceeds the number of atoms in the chain. We generated traveling solitons by giving an impulse to the atoms at the end of the chain which has free ends. However, if the end of the chain is bound to a wall, we could not find any solitary waves to sustain more than few atoms.   

Universality of Synchrony

We present a discrete model of stochastic, phase-coupled oscillators that is sufficiently simple to be characterized in complete detail, lending insight into the universal critical behavior of the corresponding nonequilibrium phase transition to macroscopic synchrony. In the mean-field limit, the model exhibits a supercritical Hopf bifurcation and global oscillatory behavior as coupling eclipses a critical value. The simplicity of our model allows us to perform the first detailed characterization of stochastic phase coupled oscillators in the locally coupled regime, where the model undergoes a continuous phase transition which remarkably displays signatures of the XY equilibrium universality class, verifying the analytical predictions of Risler et al (1). Finally, we study the model under the influence of spatial disorder and provide analytical and numerical evidence that such disorder does not destroy the capacity for synchronization. 1. T. Risler, J. Prost, F. Julicher. Phys Rev. Letters, 93 (17), (2004); Phys Rev E, 72, 016130 (2005).