Investigation of Dynamical Systems Using Symbolic Images: Efficient Implementation and Applications (original) (raw)
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Recent developments in the digital approach of symbolic dynamics
Purpose -The purpose of this paper is to explore new mathematical results to advance the understanding of the picture of a chaotic unimodal map. Design/methodology/approach -Ever since Poicare, deterministic chaos is ultimately connected with exponential divergence of nearby trajectories, unpredictability and erratic behaviour. Here, the authors propose an alternative approach in terms of complexity theory and transcendence. Findings -In this paper, the authors were able to reproduce previous results easily, due to new theorems. Originality/value -The paper updates previous results and proposes a more complete understanding of the phenomenon of deterministic chaos, also making possible connections with number theory, combinatorics and possibly quantum mechanics, as in quantum mechanics there does not exist the notion on nearby trajectories.
Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics
Physical Review Letters, 2013
We propose an algorithm for the detection of recurrence domains of complex dynamical systems from time series. Our approach exploits the characteristic checkerboard texture of recurrence domains exhibited in recurrence plots (RP). In phase space, RPs yield intersecting balls around sampling points that could be merged into cells of a phase space partition. We construct this partition by a rewriting grammar applied to the symbolic dynamics of time indices. A maximum entropy principle defines the optimal size of intersecting balls. The final application to high-dimensional brain signals yields an optimal symbolic recurrence plot revealing functional components of the signal.
Communications in Nonlinear Science and Numerical Simulation, 2013
Geometrical representations on the phase space are at the basis of Poincaré ideas of seeking structures that divide it into regions corresponding to trajectories with different dynamical fates. These ideas have been a very useful approach for studying dynamical systems. However while these representations are well achieved for autonomous and time dependent periodic dynamical systems, there is not a well established theory for describing those systems with general aperiodic time dependence. In this article we propose a methodology to build Lagrangian descriptors for arbitrary time dependent flows based on intrinsic bounded positive geometrical and physical properties of trajectories. We analyze the convenience of different descriptors from several points of view: regularity conditions requested on the vector field, rate at which the Lagrangian information is achieved and computational performance. Comparisons with other traditional methods such as FTLE are also reported.
Symbolic Computation in Nonlinear systems
IFAC Proceedings Volumes, 1997
Many mathematical concepts of linear systems have been generalized to nonlinear systems using differential geometry. We describe two problems for a wide applicability of differential geometry tools to nonlinear systems through a reliable software. One of them is the computation of the inverse of a diffeomorphism. We developed a procedure that constructs an approximate inverse on a given compact set. An example is presented where standard symbolic computation fails, and our procedure succeeds to find the approximate inverse.
2009 IEEE International Symposium on Circuits and Systems, 2009
It is shown that the problem of existence of periodic orbits can be studied rigorously by means of a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate initial positions of periodic points and interval operators are used to prove the existence of periodic orbits in a neighborhood of the computer generated solution. As an example the Lorenz system is studied. All 2536 periodic orbits of the Poincaré map with the period n ≤ 14 are found.
An Efficient Computational Framework for Studying Dynamical Systems
Proc. of the 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2013), 2013
In this paper, we introduce a computational framework for studying dynamical systems. This framework can be used to prove the existence of certain behavior in a given dynamical system at any finite (limited) resolution automatically. The proposed framework is based on approximating the phase space topology of a given dynamical system at a finite resolution by partitioning it at rational points adaptively. Dyadic rationals and partition elements with disjoint interiors are employed to build a transparent partition that enables constructing an ideal combinatorial representation of the dynamical system. Moreover, we introduce a new strategy that overcomes the sensitivity to initial conditions, supports deriving ubiquitous conclusions, allows tracking small phenomena, enables finding bifurcation points up to certain precision, and -most importantly-is computationally efficient. The new strategy is supported by a set of novel dynamic graph algorithms that are described in details. As an application, invariant sets and bifurcation points of the Logistic Map were computed.
Generalized Symbolic Dynamics Approach for Characterization of Time Series
Lecture Notes in Electrical Engineering, 2021
Various nonlinear methods have been developed to analyze the underlying dynamics of a nonlinear time series. Dynamic characterization using symbolic dynamics approach has been found to be a good alternative for the analysis of chaotic time series. As per this method, the given time series is first transformed into a single bit binary series. The single bit encoding limits its ability to capture the dynamics faithfully. This paper aims to provide a generalization of the symbolic dynamics method for better capturing the dynamical characteristics such as Lyapunov exponents of a time series. The effectiveness of the generalized method is demonstrated by employing a logistic map. The results of the analysis indicate that higher-order encoding can capture the bifurcation diagram more effectively compared to the original single bit encoding used in symbolic dynamics.
Detecting metastable states of dynamical systems by recurrence-based symbolic dynamics
We propose an algorithm for the detection of recurrence domains of complex dynamical systems from time series. Our approach exploits the characteristic checkerboard texture of recurrence domains exhibited in recurrence plots (RP). In phase space, RPs yield intersecting balls around sampling points that could be merged into cells of a phase space partition. We construct this partition by a rewriting grammar applied to the symbolic dynamics of time indices. A maximum entropy principle defines the optimal size of intersecting balls. The final application to high-dimensional brain signals yields an optimal symbolic recurrence plot revealing functional components of the signal.