Session W08: Chaos and Nonlinear Dynamics I

Enhancing synchronization by optimal correlated noise

From the flashes of fireflies to Josephson junctions and power infrastructure, networks of coupled phase oscillators provide a powerful framework to describe synchronization phenomena in many natural and engineered systems. Most real-world networks are under the influence of noisy, random inputs, potentially inhibiting synchronization. While noise is unavoidable, here we show that there exist optimal noise patterns which minimize desynchronizing effects and even enhance order. Specifically, using analytical arguments we show that in the case of a two-oscillator model, there exists a sharp transition from a regime where the optimal synchrony-enhancing noise is perfectly anti-correlated, to one where the optimal noise is correlated. We then use numerical optimization methods to find noise patterns that optimally enhance synchronization in complex oscillator networks.

Our results may have implications in real-world networks such as power grids and neuronal networks, which are subject to significant amounts of correlated input noise.   

Phase reduction theory of coherent excitable systems

Excitable systems can exhibit large excursions under superthreshold perturbations. It has been shown that fast variable noise can induce coherent oscillations in such systems due to large timescale separation, which is named as self-induced stochastic resonance (SISR). In this work, the hybrid system reproducing the SISR oscillation is established. The phase sensitivity function, which measures the phase response to infinitesimal perturbations, is calculated by solving the corresponding adjoint system. After approximated the effective frequency and noise strength, the effective phase equation is finally obtained. Our reduced model is applied to the cases of periodic forcing and two-coupled oscillators. The predicted results are in good agreement with those of Monte Carlo simulations.   

Resource Competition among Populations of Kuramoto Oscillators

The Kuramoto model has long been used to study the synchronization of populations of coupled oscillators. However, one shortcoming of the model as applied to physical systems is that true physical oscillations are not sustained indefinitely. Rather, sustained oscillations require energy. Further, the oscillators are often in competition for a limited supply of available energy. In prior work, we have extended the Kuramoto model by adding to it a resource dependence. In this work, we further allow for resource competition between the oscillators. The addition of competitive effects produces correlations between the synchronization states of different subpopulations of oscillators, for example allowing the synchronization state of one population of oscillators to be controlled only by varying the synchronization state of a distinct population, even when these two populations have no phase couplings between one another.   

Does instability imply thermalization in many-body interacting systems?

We study the relaxation process of a 1D model of interacting spins quenched far from equilibrium. Our approach is based on the direct correspondence between the classical and quantum spread of energy, initially concentrated in a particular spin-particle. The classical dynamics are quantified by the largest Lyapunov exponent, in close correspondence with the spread rate of the quantum wave packet in the many-body representation. Our results show that the initial (classical or quantum) exponential spread of energy does not necessarily imply thermalization.   

Sometimes a great motion: creating critical processes using parallel spin flips in marginally stable systems

We study the dynamics of marginally stable systems under different driving mechanisms. A typical example is the mean-field Sherrington-Kirkpatrick spin glass (SK) at T=0. The local energy minima reached, either after a thermal quench or when spins σ_i are entrained with a slowly ramping external field H_ext, exhibit a wide distribution, P(h), of their local fields h_i with whom each spin aligns. These local fields are a measure of their stability. In particular, in SK, P(h) forms a pseudo-gap, i.e., there are near-zero local fields such that P(h) ∼ h for h → 0. Thus, even minute environmental changes (like, in H_ext) will destabilize numerous spins, and SK has been shown to undergo a critical avalanche process in response. Here, we will drive the dynamics not by coupling to the spins with the external field H_ext impinging on their local fields h_i , but rather by affecting their individual stability (or “fitness”), λ_i = σ_i h_i , directly, whereas H_ext ≡ 0. Albeit somewhat unphysical, this alternate way of driving also evolves into a critical process with an interesting avalanche dynamics, which has proven to be an efficient way to find global energy minima for SK, thus, solving an NP-hard optimization problem. We will also discuss the effect of this form of driving for spin glasses without a pseudo-gap, such as the Edwards-Anderson lattice spin glass and other sparse networks.   

Many-body chaos: phase transition and noise

Chaos is an important characterisation of classical dynamical systems. How is chaos linked to the long-time dynamics of collective modes across phases and phase transitions? We address this by studying chaos across Ising and Kosterlitz-Thouless transitions in classical XXZ model. We show that spatiotemporal chaotic properties characterized by a classical out-of-time correlation (cOTOC) function have crossovers across the transitions and distinct temperature dependence in the high and low-temperature phases which show normal and anomalous diffusion, respectively. We further show that a suitably defined cOTOC can be used in an interacting system with both noise and dissipation to characterize growth and spread of chaos, which are typically defined for deterministic dynamics, unlike the non-deterministic dynamics in the presence of random noise. We show the existence of non-chaotic to chaotic transition as a function of the ratio of interaction and noise strength.   

Realizing a Gaussian Symplectic Ensemble with Quantum Spin Hall Photonic Topological Insulator Graphs

Unprecedented wave phenomena have been realized with the creation of photonic topological insulators (PTI).  One feature is the emulation of a spin-1/2 degree of freedom (DOF) in the photonic context.  The bi-anisotropic meta-waveguide (BMW) is a versatile platform for realizing analogs of quantum Hall, quantum spin Hall (QSH), and quantum valley Hall (QVH) effects for electromagnetic waves.  Random Matrix Theory has enjoyed great success in describing the statistics of large nuclei with various symmetries, and has been successfully applied to describing statistical properties of various microwave chaotic systems.  Due to the lack of a spin-1/2 DOF for light, the realization of a Gaussian Symplectic Ensemble (GSE) system was not observed for a long time in classical wave systems.

Here we aim to create a microwave simulator of GSE statistics by utilizing the intrinsic physical properties of the BMW PTIs. We conduct studies of the eigenmodes of closed PTI graphs using numerical simulations.  We construct quantum graph structures with a combination of both QSH and QVH BMW-domains, and the two-fold degeneracy of system eigenmodes are observed from eigenmode simulation results. The level spacing statistics of an ensemble of pyramidal graphs agree well with the GSE statistics.  We also present the results of an experimental realization of the QSH/QVH PTI quantum graph in superconducting Aluminum.   

Speed limits on the local stability of classical dynamical systems

Uncertainty relations are a prominent feature of quantum mechanics. However, classical systems are also characterized by a type of uncertainty – deterministic chaos – in which the uncertainty in their initial conditions leads to unpredictable behavior. In this presentation, I will discuss our theory of dynamical systems that mirrors the density matrix formulation of quantum mechanics [1]. Central to this formalism is a classical density matrix, with dynamics governed by a von Neumann-like equation of motion. and dynamical observables, such as Lyapunov exponents, that evolve in time under an Ehrenfest-like theorem. Leveraging this formalism, we derive a family of speed limits on observables in the tangent space that are set by the local dynamical (in)stability [2]. These classical speed limits are mechanical in nature and obtained from a Fisher information constructed in terms of Lyapunov vectors and the local stability matrix. For a dynamical system with a time-independent local stability matrix, these speed limits reduce to a classical analog of the Mandelstam-Tamm time-energy uncertainty relation in quantum mechanics. Our analytical and numerical results for model systems show this theory applies to arbitrary deterministic systems including those that are conservative, dissipative and driven.   

Universal Dephasing Mechanism of Many-Body Quantum Chaos

The quantum chaos conjecture states that the spectral statistics of quantum chaotic systems is universally described by the random matrix theory (RMT). It has been widely used as a diagnostics of quantum chaos, but the theoretical understanding of the underlying mechanism is far from complete. In this work, we show that the dephasing is essential for the emergence of RMT statistics. From RMT prediction, it is expected that if a generic type of interactions is introduced between the particles embedded in a chaotic medium, the exponential-in-time ramp in the spectral form factor is suppressed to a linear one, signaling a transition from single-particle to many-body chaos. We believe that this suppression is caused by the interaction-induced destruction of the quantum phase coherence between periodic paths which are related by independent time translation of each particle’s trajectory. Through a microscopic calculation of a SYK₂+SYK₂² model, we find that, in the chaotic regime, the suppression of the exponential ramp originates from a mass acquired by the diffuson-like collective modes - a manifestation of the dephasing effect due to interactions. In the integrable regime, these collective modes remain massless in the presence of interactions, suggesting the failure of dephasing.   

Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction

Entrainment of self-sustained oscillators by periodic inputs is widely observed in the real world, including the entrainment of circadian rhythms to sunlight and injection locking of electrical oscillators to clock signals. When the perturbation given to the oscillator is sufficiently weak, the phase reduction theory can be employed to approximate the multidimensional nonlinear dynamics of the oscillator by a one-dimensional phase equation. Recently, optimization of periodic inputs has been studied by using the phase equation to realize fast entrainment and high energy efficiency. However, this method does not work well for strong inputs because the amplitude deviations of the oscillator state from the original orbit become large, and the phase-only approximation of the system breaks down. In this study, using phase-amplitude reduction, we propose two methods to find the input waveforms that can suppress the amplitude deviations even for strong inputs. We demonstrate that the proposed method allows us to use stronger inputs, thereby achieving faster convergence to the target phase-locking point than the conventional method based only on the phase equation.   

Proportional selective modification of dynamical systems by "exterior dissipation"

I will introduce a technique for adding dissipation or otherwise modifying dynamical systems to selectively change any number of conserved quantities, while only reducing the total number of conserved quantities by one.  I will first present a naïve approach to a simple example, a textbook problem of a specially damped rotor often used to explain the famous failure of the Explorer 1 satellite.  Then (in joint work with M. Aureli), we generalize the approach to any number of dimensions and conserved quantities.  The resulting dynamics drives the modified system to a nontrivial state of the original system, such as a limit cycle.