Investigation of Dynamical Systems Using Symbolic Images: Efficient Implementation and Applications

Abstract

Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this graph as well as attractors, their basins and connecting orbits. Extensions of the framework allow, e.g. the calculation of the Morse spectrum and verification of hyperbolicity. In this work, efficient algorithms and adequate data structures will be presented for the construction of symbolic images and some basic operations on them, like the localization of the chain recurrent set and periodic orbits. The performance of these algorithms will be analyzed and we show their application in practice. The focus is not only put on several standard systems, like Lorenz and Ikeda, but also on some less well-known ones. Additionally, some tuning techniques are presented for an efficient usage of the method.

Doubling bifurcation of a closed invariant curve in 3D maps

“…On the other hand, when it is associated with cycles of a high period, the period doubling bifurcations of the two cycles occur almost simultaneously, so that the effect can be numerically observed, but it is difficult to detect the related period. 1 We thank Soumitro Banerjee for this figure. Clearly the node becomes a saddle, and an attracting node of double period C 2N appears, with periodic points on both sides of the unstable manifold issuing from C N . Similarly for the saddle, a second branch becomes unstable, and a saddle of double period C 2S appears, with periodic points on both sides of the unstable manifold issuing from C S .…”

“…In several applied models (see, e.g., [1,[3][4][5]) it has been observed that closed invariant curves in three dimensional (3D for short) maps can undergo a kind of doubling bifurcation sequence. The bifurcation mechanism leading to such a dynamic behavior was announced as an open problem at the European Conference on Iteration Theory in 2006 (ECIT-06) [6].…”

“…As an example, in Fig.1 we show the closed invariant attracting curve and a sequence of the bifurcations mentioned above in the Lotka-Volterra model represented by the 3D map x = x + Rx(1 − x − ay − bz) y = y + Ry(1 − bx − y − az) z = z + Rz(1 − ax − by − z) (1) where the parameters R and a are fixed as R = 1, a = 0.5, and b is varied (taken from [5] and [3]). Our aim is to give a qualitative description of the possible mechanism of such bifurcations.…”

Onset of chaos in a single-phase power electronic inverter

“…Unstable cycles are detected by the variant of the cell-to-cell mapping method 14,15 described in Ref. 16. Multiplier q of the fixed point is calculated by Eq.…”

Flows on graphs and dynamical systems

“…The symbolic image G is a directed graph such that vertex i corresponds to the cell M (i) and the edge i → j exists if f (M (i))∩M (j) = ∅. The notion of symbolic image of a dynamical system was introduced in 1983 [9], monograph [10] and paper [1] describe the symbolic image methods and their applications. We will use symbolic image construction with respect to a partition C. In general case, there are no any restrictions on properties of cells.…”