Low-dimensional geometry—A variational approach

2006 ◽  
pp. 183-207
Author(s):  
Nigel Hitchin
Keyword(s):  
Variational Approach ◽  
Low Dimensional

Periodic orbits analysis for the Zhou system via variational approach

2019 ◽  
Vol 33 (19) ◽  
pp. 1950212 ◽  
Author(s):  
Chengwei Dong ◽  
Lian Jia
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Variational Approach ◽  
Topological Classification ◽  
Dissipative Systems ◽  
Systematic Calculation ◽  
Low Dimensional ◽  
General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

scholarly journals Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenchong Tian ◽  
Hao Wu
Keyword(s):  
Dynamical Systems ◽  
Variational Approach ◽  
Distance Measure ◽  
Transfer Operator ◽  
State Variables ◽  
Modeling And Analysis ◽  
Koopman Operator ◽  
Linear Representations ◽  
Dimensional Approximation ◽  
Low Dimensional

Abstract Transfer operators such as Perron–Frobenius and Koopman operator play a key role in modeling and analysis of complex dynamical systems, which allow linear representations of nonlinear dynamics by transforming the original state variables to feature spaces. However, it remains challenging to identify the optimal low-dimensional feature mappings from data. The variational approach for Markov processes (VAMP) provides a comprehensive framework for the evaluation and optimization of feature mappings based on the variational estimation of modeling errors, but it still suffers from a flawed assumption on the transfer operator and therefore sometimes fails to capture the essential structure of system dynamics. In this paper, we develop a powerful alternative to VAMP, called kernel embedding based variational approach for dynamical systems (KVAD). By using the distance measure of functions in the kernel embedding space, KVAD effectively overcomes theoretical and practical limitations of VAMP. In addition, we develop a data-driven KVAD algorithm for seeking the ideal feature mapping within a subspace spanned by given basis functions, and numerical experiments show that the proposed algorithm can significantly improve the modeling accuracy compared to VAMP.

Unstable cycles for the Burke–Shaw system via variational approach

2019 ◽  
Vol 33 (21) ◽  
pp. 1950240
Author(s):  
Chengwei Dong ◽  
Huihui Liu
Keyword(s):  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Topological Structure ◽  
Variational Approach ◽  
Chaotic Systems ◽  
Dissipative System ◽  
Section Method ◽  
Varied Structure ◽  
Low Dimensional

In this paper, the systematical calculations of the unstable cycles for the Burke–Shaw system (BSS) are presented. In contrast to the Poincaré section method used in previous studies, we used the variational method for the cycle search and established appropriate symbolic dynamics on the basis of the topological structure of the cycles. The variational approach made it easy to continuously track the periodic orbits when the parameters were varied. Structure of the whole cycle in the dissipative system demonstrated that the methodology could be effective in most low-dimensional chaotic systems.

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