PHYSICAL REVIEW E 84, 046205 (2011) Nonuniqueness of global modeling and time scaling Claudia Lainscsek The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA and Institute for Neural Computation, University of California at San Diego, La Jolla, California 92093, USA (Received 26 August 2010; revised manuscript received 19 September 2011; published 12 October 2011) Starting from an observed single time series, it is shown how to reconstruct a global model in the original phase space by using the ansatz library approach. This model is then compared to the underlying dynamical system that describes the initial time series, and the nonuniqueness of the reconstructed model is discussed. This framework is extended by taking an additional time scaling factor in the reconstructed model class under consideration. DOI: 10.1103/PhysRevE.84.046205 PACS number(s): 05.45.−a I. INTRODUCTION To study an mth order dynamical system, all m physical quantities should be known to have a complete description of the system under investigation. In most experimental situations, only a single quantity can be measured. The embedding theorem of Takens [1] shows us how to get insight into the whole dynamical system from this incomplete set of measurements. Examples of modeling of physical experiments on chaotic systems include chemical reactions [2,3], vibrating strings [4], optical fiber ring resonators [5], and laser and sunspot data [6], among others. There are two main model types: the phenomenological models that need specialized knowledge about the system under study and models that are based on the time series data, which are the subject of this paper. Time series based modeling can yield linear stochastic models (e.g. AutoRegressive Moving Average models) or deterministic models (local or global). There are several different types of global deterministic models, such as neural networks or differential equation models, and the basis functions of these models can be, e.g., polynomials or radial basis functions. A nice data driven introduction to all these techniques can be found in [7], which is based on the freely available software package TISEAN [8]. Another good review of modeling techniques (local and global, linear and nonlinear) can be found in [9]. In this paper, global nonlinear deterministic ordinary differential equations (ODE) models with polynomials will be considered. Global modeling strongly depends on the time series available, and the papers of Letellier and Aguirre [10,11] explain how the choice of observables influences the amount of information we can achieve from such a single time series of a nonlinear dynamical system. Reconstruction of a system of ODEs from a single time series then can be done using differential coordinates [12,13]. The ansatz library [13–15] can then further be used to reconstruct a dynamical system in the original phase space. The problems that have been neglected in these papers are the questions on uniqueness of the reconstructed model and the role of time scaling in the time series under consideration. In this paper, it is assumed that a time series is available and the following questions are asked: (1) Is it possible to create a model, within a certain class of models, that describes the initial time series? (2) Is the model unique? If not, what is the degree of nonuniqueness within the class of models treated? (3) What is the role of time scaling? To answer these questions, the ansatz library approach [13] is used where a three-dimensional (3D) system of ODEs is proposed that can be converted to jerk form with polynomial functions. The transformation between the original dynamical system and the jerk or differential model provides a relation between the parameters of these two systems. When a differential model is extracted from a scalar time series, this relation of the parameters can be used to extract the coefficients and model form in the initial model. This idea is illustrated with the example of the R¨ ossler system. The paper is organized as follows: In Sec. II, the trans- formation of a dynamical system to its differential model is introduced. This transformation is discussed, and it is shown that there exists a whole class of dynamical systems that share the same differential model. In Sec. III, this is extended to an additional time scaling factor. Section IV is the conclusion. II. GENERAL DESCRIPTION OF THE PROBLEM A. Background The class of models considered here is a 3D system of ODEs with the right-hand sides containing polynomials with up to second order nonlinearities, which can be written in a general form as ˙ x i = a i,0 + a i,1 x 1 + a i,2 x 2 + a i,3 x 3 + a i,4 x 2 1 + a i,5 x 1 x 2 + a i,6 x 1 x 3 + a i,7 x 2 2 + a i,8 x 2 x 3 + a i,9 x 2 3 , i = 1,2,3. (1) Usually, only a small subset of coefficients a i,∗ is assumed to be nonzero [13]. This subset defines the class of models under consideration. This class has N m nonzero parameters a i,∗ . 1. R¨ ossler-type models We ask the following question: Is it possible that the time series corresponds to a function of the variables φ(x i )? For simplicity, does it correspond to one of the variables x i ? To answer these questions, the x 2 variable of the R¨ ossler system [16] ˙ x 1 = a 1,2 x 2 + a 1,3 x 3 , ˙ x 2 = a 2,1 x 1 + a 2,2 x 2 , (2) ˙ x 3 = a 3,0 + a 3,3 x 3 + a 3,6 x 1 x 3 046205-1 1539-3755/2011/84(4)/046205(4) ©2011 American Physical Society