Chaos

We use recent advances in the machine learning area known as 'reservoir computing' to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a 'reservoir'. After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the 'output weights'. The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing arbitrarily long time series whose ergodic properties approximate those of the input signal. When successful, we say that the autonomous reservoir reproduces the attractor's 'climate'. Since the reservoir equations and output weights are known, we can compute derivatives needed to determine the Lyapunov exponents of the autonomous reservoir, which we then use as estimates of the Lyapunov exponents for the original input generating system. We illustrate the effectiveness of our technique with two examples, the Lorenz system, and the Kuramoto-Sivashinsky (KS) equation. In particular, we use the Lorenz system to show that achieving climate reproduction may require tuning of the reservoir parameters. For the case of the KS equation, we note that as the system's spatial size is increased, the number of Lyapunov exponents increases, thus yielding a challenging test of our method, which we find the method successfully passes.

Jaideep Pathak

Zhixin Lu

Brian R. Hunt

Michelle Girvan

Edward Ott

Chaos An Interdisciplinary Journal of Nonlinear Science

Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data

When a system of ordinary differential equations is discontinuous along some threshold, its flow may become tangent to that threshold from one side or the other, creating a fold singularity, or from both sides simultaneously, creating a two-fold singularity. The classic two-fold exhibits intricate local dynamics and accumulating sequences of local bifurcations and is by now rather well understood, but it is just the simplest of an infinite hierarchy of two-folds and multi-folds in higher dimensions. These arise when a system is discontinuous along multiple intersecting thresholds, and the induced sliding flows on those thresholds become tangent to their intersections. We show here, surprisingly, that these higher dimensional analogs of the two-fold reduce to the equations of the classic two-fold, providing the first step into their study and a new tool to understand higher dimensional systems with discontinuities.

Simon Webber

Mike R. Jeffrey

Two-fold singularities in nonsmooth dynamics—Higher dimensional analogs

非光滑动力学中的二重奇异性——高维类比

Consider the generic family of 3D Filippov linear systems that possess a double-tangency singularity of Teixeira type. We are interested in finding mechanisms for the emergence of an attractor from such a singularity, like a crossing limit cycle, an invariant torus, or a strange attractor. For this, we unfold the pseudo-Hopf bifurcation for this class of systems in order to guarantee the existence of a crossing limit cycle and, subsequently, from this attractor, obtain a more intricate one. Two illustrative examples are given in order to show evidence of attractors obtained by means of the proposed strategy. Both theoretical and numerical results are provided for verification and demonstration.

Juan Castillo

The pseudo-Hopf bifurcation and derived attractors in 3D Filippov linear systems with a Teixeira singularity

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

Igor Belykh

Rachel Kuske

Maurizio Porfiri

David J. W. Simpson

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

In this paper, we propose a novel control approach for opinion dynamics on evolving networks. The controls modify the strength of connections in the network, rather than influencing opinions directly, with the overall goal of steering the population toward a target opinion. This requires that the social network remains sufficiently connected, the population does not break into separate opinion clusters, and that the target opinion remains accessible. We present several approaches to address these challenges, considering questions of controllability, instantaneous control, and optimal control. Each of these approaches provides a different view on the complex relationship between opinion and network dynamics and raises interesting questions for future research.

A. Nugent

S. N. Gomes

M. T. Wolfram

Steering opinion dynamics through control of social networks

通过控制社交网络引导舆论动态

In many applications, one is interested in classifying trajectories of symplectic maps as invariant tori, islands, or chaos. Here, we introduce Birkhoff reduced rank extrapolation (RRE), a modified version of the RRE method, which efficiently obtains ergodic averages of invariant tori and islands in symplectic maps. In contrast, Birkhoff RRE does not efficiently converge in chaos, meaning its convergence rate can be used for trajectory classification. Furthermore, for the islands and invariant circles, Birkhoff RRE returns the number of islands and the rotation number. This allows us to find Fourier parameterizations of invariant circles and islands from a single trajectory. We show examples of Birkhoff RRE on the standard map and magnetic field line dynamics.

M. Ruth

D. Bindel

Finding Birkhoff averages via adaptive filtering

通过自适应滤波寻找Birkhoff平均

We propose two thermodynamically consistent inertial spin models generalizing the inertial spin model for the ensemble of mechanical particles with spin. For the collective dynamics modeling consistent with an entropy principle, we use two modeling ideas based on Hamiltonian formulation and flocking formalism for the coupling of the inertial spin model and temperatures introduced by Ha and Ruggeri [Arch. Ration. Mech. Anal. 223, 1397 (2017)]. The resulting models admit several constants of motions and are consistent with the entropy principle. We present several sufficient frameworks leading to the asymptotic vanishing of spins and velocity alignment. We also implement several numerical simulations and compare them with analytical results.

Seung-Yeal Ha

Guanghui Jin

Youwei Ma

Wook Yoon

Emergent behaviors of two thermodynamically consistent inertial spin models

两个热力学一致性惯性自旋模型的涌现行为

This study investigates the most probable trajectories of a birhythmic oscillator under stochastic perturbations. The distinctive feature of the birhythmic oscillator is the coexistence of two stable limit cycles with different amplitudes and frequencies, separated by an unstable limit cycle. The path integral method was utilized to compute the instantaneous probability density. Based on the theory of most probable dynamics, by maximizing the probability density function, we present the time series of the most probable trajectories starting from different initial states. Furthermore, we conducted a detailed analysis of the noise-induced transitions between the two stable limit cycles under different parameter conditions. This approach enables us to understand and track the most probable escape time and specific most probable trajectories as the system transitions from the basin of attraction of one stable limit cycle to another. This work visualizes the most probable trajectories in stochastic systems and provides an innovative solution to the complex problem of noise-induced transitions between two stable limit cycles. Our research aims to provide a new perspective for studying complex stochastic dynamical systems.

Wenting Zhang

Wei Xu

Yaning Tang

Jürgen Kurths

Most probable trajectories of a birhythmic oscillator under random perturbations

随机扰动下双节律振荡器的最可能轨迹

Using methods of numerical simulation, we demonstrate the constructive role of memristive coupling in the context of the traveling wave formation and robustness in an ensemble of excitable oscillators described by the FitzHugh–Nagumo neuron model. First, the revealed aspects of the memristive coupling action are shown in an example of the deterministic model where the memristive properties of the coupling elements provide for achieving traveling waves at lower coupling strength as compared to non-adaptive diffusive coupling. In the presence of noise, the positive role of memristive coupling is manifested as significant, increasing a noise intensity critical value corresponding to the noise-induced destruction of traveling waves as compared to classical diffusive interaction. In addition, we point out the second constructive factor, the Lévy noise, whose properties provide for inducing traveling waves.

Ivan A. Korneev

Ibadulla R. Ramazanov

Andrei V. Slepnev

Tatiana E. Vadivasova

Vladimir V. Semenov

Traveling waves in an ensemble of excitable oscillators: The interplay of memristive coupling and noise

兴奋振子集合中的行波：忆阻耦合与噪声的相互作用

The framework of mean-field games (MFGs) is used for modeling the collective dynamics of large populations of non-cooperative decision-making agents. We formulate and analyze a kinetic MFG model for an interacting system of non-cooperative motile agents with inertial dynamics and finite-range interactions, where each agent is minimizing a biologically inspired cost function. By analyzing the associated coupled forward–backward in a time system of nonlinear Fokker–Planck and Hamilton–Jacobi–Bellman equations, we obtain conditions for closed-loop linear stability of the spatially homogeneous MFG equilibrium that corresponds to an ordered state with non-zero mean speed. Using a combination of analysis and numerical simulations, we show that when energetic cost of control is reduced below a critical value, this equilibrium loses stability, and the system transitions to a traveling wave solution. Our work provides a game-theoretic perspective to the problem of collective motion in non-equilibrium biological and bio-inspired systems.

Piyush Grover

Mandy Huo

Phase transition in a kinetic mean-field game model of inertial self-propelled agents

惯性自推进代理的动量场博弈模型中的相变

acBWQ6	Chaos
Journal Impact	2.67
mbqgSu	PHYSICS, MATHEMATICAL(Q1)
xGGr6y	SCI Q1中科院 2 区

ISSN	1054-1500, 1089-7682, 1527-2443
h-index	127
hQoIdU	数学(2区)数学物理数学物理(2区)数学应用数学(3区)

u9zg3I	物理数学计算机科学量子力学人工智能工程类数学分析经济生物统计物理学非线性系统管理热力学组合数学电气工程统计化学几何学混乱的材料科学电信控制（管理）
+5d/bs	e+/GRg: American Institute of Physics，uzYam8: Quarterly，dxwaWB: journal，8x2mMQ: TY4X7V
ak9yM6	A7IHrH: 1991，sQJ/D8: 98.21%，sE9og+:18.50%， wzlKC/: 15.12%
JrpmBe	网友分享经验：平均3.0个月
jdemjX	网友分享经验：较难

Chaos

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Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data

Two-fold singularities in nonsmooth dynamics—Higher dimensional analogs

The pseudo-Hopf bifurcation and derived attractors in 3D Filippov linear systems with a Teixeira singularity

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

Steering opinion dynamics through control of social networks

Finding Birkhoff averages via adaptive filtering

Emergent behaviors of two thermodynamically consistent inertial spin models

Most probable trajectories of a birhythmic oscillator under random perturbations

Traveling waves in an ensemble of excitable oscillators: The interplay of memristive coupling and noise

Phase transition in a kinetic mean-field game model of inertial self-propelled agents