Modular chaos for random processes

Domain structured chaos for discrete random processes

We introduce the notion of domain-structured chaos and apply it to establish a connection between stochastic dynamics and deterministic chaos.

General definitions of chaos for continuous and discrete-time processes

A precise definition of chaos for discrete processes based on iteration already exists. We shall first reformulate it in a more general frame, taking into account the fact that discrete chaotic behavior is neither necessarily based on iteration nor strictly related to compact metric spaces or to bounded functions. Then we shall apply the central idea of this definition to continuous processes. We shall try to see what chaos is, regardless of the way it is generated.

Chaos in Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations ${u_t}$ = ${z_t}$ $(1-\sum\limits_{j=1}^q β_j L^j)σ_{t}^2 = ω+(1-\sum\limits_{j=1}^q β_j L^j - (\sum\limits_{k=1}^p φ_k L^k) (1-L)^d) u_t^2$, where $ω\in$ R, and $β_j\in$ R are constant parameters, $\{u_t\}_{{t\in}^+}$ and $\{σ_t\}_{{t\in}^+}$ are the discrete time real valued stochastic processes which represent FIGARCH (p,d,q) and stochastic volatility, respectively. Mo…

弱混沌系统的统计特性

Statistics of Weakly Chaotic Systems

本文综述弱混沌系统的统计理论，介绍其背景、与强混沌的区别及相关新结果，该理论因问题复杂发展程度低于强混沌理论。

Detecting chaos in lineage-trees: A deep learning approach

Many complex phenomena, from weather systems to heartbeat rhythm patterns, are effectively modeled as low-dimensional dynamical systems. Such systems may behave chaotically under certain conditions, and so the ability to detect chaos based on empirical measurement is an important step in characterizing and predicting these processes. Classifying a system as chaotic usually requires estimating its largest Lyapunov exponent, which quantifies the average rate of convergence or divergence of initially close trajectories in state space, and for which a positive value is generally accepted as an ope…