Autonomous learning by simple dynamical systems with a discrete-time formulation

We present a discrete-time formulation for the autonomous learning conjecture. The main feature of this formulation is the possibility to apply the autonomous learning scheme to systems in which the errors with respect to target functions are not well-defined for all times. This restriction for the evaluation of functionality is a typical feature in systems that need a finite time interval to process a unit piece of information. We illustrate its application on an artificial neural network with feed-forward architecture for classification and a phase oscillator system with synchronization properties. The main characteristics of the discrete-time formulation are shown by constructing these systems with predefined functions.     

Chaotification of discrete dynamical systems via impulsive control

This Letter is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via impulsive control techniques. Chaotification theorems for one-dimensional discrete dynamical systems and general higher-dimensional discrete dynamical systems are derived, respectively, whether the original systems are stable or not. Finally, the effectiveness of the theoretical results is illustrated by some numerical examples.     

Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.     

Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems

In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.     

Synchronization of Rossler and Chen chaotic dynamical systems using active control

This Letter presents chaos synchronization of two identical Rossler and Chen systems by using active control. The proposed technique is applied to achieve chaos synchronization for the Rossler and Chen dynamical systems. We demonstrate that a coupled Rossler and Chen systems can be synchronized. Numerical simulations are used to show the effectiveness of the proposed control method.     

Universal dynamical control of local decoherence for multipartite and multilevel systems

A unified theory is given of dynamically modified decay and decoherence of field-driven multilevel multipartite entangled states that are weakly coupled to zero-temperature baths or undergo random phase fluctuations. The theory allows for arbitrary local differences in their coupling to the environment. Due to such differences, the optimal driving-field modulation to ensure maximal fidelity is found to substantially differ from conventional “Bang-Bang” or π-phase flips of the single-qubit evolution.     

Iterated function systems and dynamical systems

We study the relationship between measures invariant for a piecewise expanding transformation tau of a compact metric space endowed with a underlying measure and measures invariant for an iterated function system T(tau), generated by inverse branches of tau. The main result says that the tau-invariant absolutely continuous measure &mgr; is also T(tau) invariant if and only if tau is absolutely continuously conjugated with a piecewise linear transformation. Measures of maximal entropy and general equilibrium states are also discussed. (c) 1995 American Institute of Physics.     

Punctures and dynamical systems

Letters in Mathematical Physics - With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of...     

Complex network perspective on modelling chaotic systems via machine learning

Recent advances have demonstrated that a machine learning technique known as "reservoir computing" is a significantly effective method for modelling chaotic systems. Going beyond short-term prediction, we show that long-term behaviors of an observed chaotic system are also preserved in the trained reservoir system by virtue of network measurements. Specifically, we find that a broad range of network statistics induced from the trained reservoir system is nearly identical with that of a learned chaotic system of interest. Moreover, we show that network measurements of the trained reservoir system are sensitive to distinct dynamics and can in turn detect the dynamical transitions in complex systems. Our findings further support that rather than dynamical equations, reservoir computing approach in fact provides an alternative way for modelling chaotic systems.     

Data-driven modeling and parameter estimation of nonlinear systems

The European Physical Journal B - The lattice parameters, formation energy and electronic structure as well as elastic property and Debye temperature of Ca-doped Ti2Ni alloy have been calculated....     

Localization in disordered,nonlinear dynamical systems

We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrödinger equations, or randomly coupled rotors.     

Synchronization engineering: theoretical framework and application to dynamical clustering

A method for engineering the global behavior of populations of rhythmic elements is presented. The framework, which is based on phase models, allows a nonlinear time-delayed global feedback signal to be constructed which produces an interaction function corresponding to the desired behavior of the system. It is shown theoretically and confirmed in numerical simulations that a polynomial, delayed feedback is a versatile tool to tune synchronization patterns. Dynamical states consisting of one to four clusters were engineered to demonstrate the application of synchronization engineering in an experimental electrochemical system.     

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.     

Book review: Nonlinear oscillations,dynamical systems,and bifurcations of vector fields

Journal of Statistical Physics -     

Autonomous systems,dynamical systems,LPTI symmetries,topology of trajectories,and periodic solutions

In the case of autonomous dynamical systems, it is better to base symmetry considerations on trajectories than on full solutions. In this setting topological arguments can be used; a special role is played in this context by time-independent Lie-point symmetries. As an application of this approach, we obtain results on the existence of stationary and/or periodic solutions.     

“电路原理”、“信号与系统”和“自动控制原理”课程的整合与优化

"电路原理"、"信号与系统"和"自动控制原理"三门课程是光电信息科学与工程等专业的专业基础课。若分别采用各自独立的教学内容和方法,内容交叉、重复,浪费学时,没有从整个课程体系系统化的角度把握有关内容。因此,提出了适用于光电信息科学与工程等专业的以上三门课程的教学改革方案,从根本上系统地将这三门课程进行整合、优化为"电路、信号和控制系统"课程。在教学实践中,进行了有关内容的教学尝试,缩短了教学时间,提高了教学质量。     

Hyperbolification of dynamical systems: The case of continuous-time systems

We present a new method to generate chaotic hyperbolic systems. The method is based on the knowledge of a chaotic hyperbolic system and the use of a synchronization technique. This procedure is called hyperbolification of dynamical systems. The aim of this process is to create or enhance the hyperbolicity of a dynamical system. In other words, hyperbolification of dynamical systems produces chaotic hyperbolic (structurally stable) behaviors in a system that would not otherwise be hyperbolic. The method of hyperbolification can be outlined as follows. We consider a known n-dimensional hyperbolic chaotic system as a drive system and another n-dimensional system as the response system plus a feedback control function to be determined in accordance with a specific synchronization criterion. We then consider the error system and apply a synchronization method, and find sufficient conditions for the errors to converge to zero and hence the synchronization between the two systems to be established. This means that we construct a 2n-dimensional continuous-time system that displays a robust hyperbolic chaotic attractor. An illustrative example is given to show the effectiveness of the proposed hyperbolification method.