Evaluation of dynamical models: Dissipative synchronization and other techniques Luis Antonio Aguirre, * Edgar Campos Furtado, and Leonardo A. B. Tôrres Laboratório de Modelagem, Análise e Controle de Sistemas Não-Lineares, Programa de Pós-Graduação em Engenharia Elétrica, Universidade Federal de Minas Gerais, Avenue Antônio Carlos 6627, 31270-901 Belo Horizonte, Minas Gerais, Brazil Received 15 September 2005; revised manuscript received 28 August 2006; published 13 December 2006 Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should repro- duce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminat- ing power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing- Holmes, and van der Pol oscillators, plus the Hénon map. The procedure, developed for discrete-time systems, is focused on the dynamical properties of the model, rather than on statistical issues. For all the systems investigated, it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams—which in turn is much greater than, say, that of correlation dimension—but at a much lower computational cost. DOI: 10.1103/PhysRevE.74.066203 PACS numbers: 05.45.Tp, 05.45.Xt, 07.05.Tp, 07.05.Kf I. INTRODUCTION Model building from data has been of great interest for many years within the community of nonlinear dynamics since one of the first works in this field 1. For the last 20 years or so, many different procedures have been put forward for building nonlinear models from data. In a sense, the field of model building is now rather mature within the commu- nity of nonlinear dynamics. After building a model it is important to know if such a model is in fact a dynamical analog of the original system. An answer to that question is searched for during model validation. Among the several issues concerning model building, model validation is probably the one that has re- ceived the least attention. For instance, in 2 an interesting discussion of several aspects of global modeling is found; however, model validation is hardly mentioned. Many of the tools for model validation that were com- monly used by the mid-1990s were investigated and com- pared in 3. The main conclusion of such a work was that if a global model of a system is required, then one of the most exacting procedures for model validation is to compare the model bifurcation diagram to that of the true system. Two similar diagrams 1 point to two entities usually system and model which display the same dynamical regimes over a rather wide range of parameter values. In fact it is widely acknowledged that bifurcation diagrams “are one of the most informative forms of presentation of dynamical evolution” 4. On the other hand, quantities such as dimension mea- sures, Lyapunov exponents, phase portraits, and so on can only quantify attractors and actually say very little about the model ability to mimic the system as it evolves from one dynamical regime attractor to another. Thus, good models should match closely such geometrical invariants; however, that matching is insufficient on its own to guarantee the qual- ity of such models 5. In fact, it is known that even two drastically different attractors may have similar fractal di- mension or Lyapunov exponents. Although bifurcation diagrams provide a very exacting means of verifying the dynamical overall behavior of a model, their practical use is somewhat limited to those cases in which it is viable to obtain such a diagram for the original system. 2 Another practical difficulty is that model validation using bifurcation diagrams is generally quite subjective, as will be illustrated in Sec. IV. In addition, the numerical de- termination of bifurcation diagrams could become rather de- manding. In order to overcome such shortcomings of the bifurcation diagrams as a tool for model selection, this paper proposes a way of choosing from a set of candidate models based on dissipative synchronization. To assess the perfor- mance of the our method, bifurcation diagrams are used be- cause they are known to be a hard test when the model dy- namics are in view. Having said that, it is worth pointing out that bifurcation diagrams have been used in model validation in several contexts 6–16, where in some cases the systems were autonomous i.e., had no, time-dependent, exogenous variables. The remainder of the paper is organized as follows. Sec- tion II surveys some of the most commonly used methods of model validation applied to nonlinear dynamics. In that sec- tion three rather recent developments are mentioned in some detail. In Sec. III a procedure for model evaluation is pre- sented. This procedure is based on dissipative synchroniza- tion. The ideas are tested using three benchmark systems, and the performance of the synchronization scheme is com- pared to that of other methods in Sec. IV. The main conclu- sions of the paper are provided in Sec. V. *Corresponding author. FAX: +55 31 3499-4850. Email address: aguirre@cpdee.ufmg.br 1 By similar it is meant that both model and system undergo the same sequence of bifurcations and display the same dynamical re- gimes for approximately the same values of the bifurcation parameter. 2 A very interesting example has been published recently by Small and coworkers who have discussed the estimation of a bifurcation diagram from a set of biomedical data 77. PHYSICAL REVIEW E 74, 066203 2006 1539-3755/2006/746/06620316 ©2006 The American Physical Society 066203-1