Estimation of periodic-like motions of chaotic evolutions using detected unstable periodic patterns
Introduction
Estimation of chaotic behaviors is one of the challenging topics in the prediction of dynamical systems. A chaotic process is deterministic rather than random, such that the variation of the chaotic system may be predictable in the short term based on the deterministic property. Deterministic models could be constructed based on state space reconstruction from time series with delay coordinates (Takens, 1981). In deterministic approach, local linear approximations have been widely adopted for the estimation of evolutions based on optimal metrics (Garcia et al., 1996) and ordinary least squares (Kugiumtzis et al., 1998) as examples. The deterministic approach is a localized linear method. It omits useful information of the inherent cyclical patterns that are embedded within chaotic data.
In this paper, we introduce a pattern-based approach to estimate the evolution of chaotic motions by making use of unstable periodic orbits (UPOs) in chaos. The method is developed based on the following nature of chaos. An initially chaotic motion may suddenly be tamed by a UPO, and appear to be a periodic-like mode similar to the UPO pattern. Such a periodic-like motion may persist for a while and will eventually revert back to the irregular behavior due to the unstable nature of the UPO. Occurrence of many different types of periodic-like motions in a chaotic evolutionary course is subjected to the existence of a number of UPOs resided in chaos (Grebogi et al., 1987). A chaotic trajectory may simulate the pattern of any of the UPOs as it is nearby the UPO. Thus UPO patterns act as road marks which can be used to estimate cyclical motions in chaotic evolutions.
Recognition of cyclical patterns and estimation of complex dynamics may lead to many applications, for instance, business cycle detection, seasonal changes in meteorology and population variations in ecology. The paper is arranged as follows. A pattern recognition method is introduced to extract UPO patterns from chaotic data in Section 2. Then we elaborate the idea of how to use UPO patterns for developing an estimation model in Section 3. A numerical experiment is illustrated in 4 Chaotic motion of a business cycle model, 5 Pattern detection and motion estimation to detect and estimate business cycles and the conclusion is given in Section 6.
Section snippets
Extraction of UPOs patterns in chaos
UPOs are inherent periodic patterns resided within a chaotic attractor and can be detected from chaotic data. Consider an n-dimensional system (continuous or discrete-time system) which can be expressed by the mapwhere is a mapping point at the ith time step and an n×1 vector, behaving in a chaotic fashion.
When evolves close to in m time steps (close return pair), it implies that there exists a period-m UPO nearby the two mapping points. The UPO path is
Trend estimation using UPO patterns
With the objective of estimating the cyclical behavior of chaotic systems, we can actually set up a capture region around any point of the detected UPOs path denoted by , as displayed in Fig. 1. Place a number of monitoring sections Γk (k=1,2,3,…) in a phase space with each in a time interval τ. Once the chaotic trajectory enters the capture region (due to the property of ergodicity of chaos), the trajectory will follow the detected UPOs pattern closely. We can thus project the
Chaotic motion of a business cycle model
The Kaldor business cycle model (Kaldor, 1940) is one of the renowned economic models, and involves nonlinear investment and saving functions that shift time in response to capital accumulation or decumulation. Serving as a prototype model for nonlinear dynamical systems in economics, the Kaldor model has received much research interest (Lorenz, 1993; Gabisch and Lorenz, 1989). The Kaldor model in two-dimensional discrete-time form can be written as
Pattern detection and motion estimation
Detection of UPOs is a process of the pattern recognition. We are able to obtain the patterns of UPOs by employing the locating method introduced in Section 2. In the numerical experiment, the locations of a number of UPOs are detected from the chaotic time series. Here a period-10 UPO is displayed in Fig. 4.
With the recognized UPO patterns, we can perform trend estimation of chaotic evolution by applying the proposed method in Section 3. The UPO displayed in Fig. 4 is selected for the
Conclusions
We put forward the idea that UPOs can be exploited to predict periodic-like motions that often appear in chaotic dynamics. The UPOs are hidden cyclic patterns that are embedded within a chaotic attractor. These patterns form a skeleton structure expanded over a phase space where a chaotic attractor settles. UPO patterns play an important role in dominating the dynamical evolution of chaos such that a chaotic trajectory will be `forced' to move along the pattern of a UPO for a while as it is
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