Chemical Engineering Science 56 (2001) 5643–5651 www.elsevier.com/locate/ces Characterization of chaotic dynamics—II: topological invariants and their equivalence for an autocatalytic model system and an experimental sheared polymer solution S. Deshmukh, A. Ghosh, M. V. Badiger, V. Ravi Kumar, B. D. Kulkarni * Chemical Engineering Division, National Chemical Laboratory, Pune 411 008, India Abstract Characterization of strange attractors exhibiting chaotic dynamics may be carried out through computation of metric, dynamical and topological invariants. The last of these are robust even under control parameter variations and hence have certain distinct advantages. In the present work, we carry out the topological analysis of the observed dynamics from a model autocatalytic reacting system and an experimental polymer solution subjected to shear. Low dimensional chaotic dynamics are observed in both these systems. The results show the global characterization and classication of the dynamics for both systems based on topological invariants, viz., linking numbers and relative rotational rates, is possible. The analyses of these invariants yield the template and the Markov transition matrix that contain in them valuable topological information about the system dynamics. The results obtained show that the two systems possess similar topological characteristics and follow the horseshoe mechanism. This information should help in developing design and control algorithms for these systems. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Topological invariants; Chaos; Nonlinear dynamics; Autocatalysis; Sheared polymer solution 1. Introduction Chaotic dynamical systems have been the concern of numerous studies and the eort has provided signicant insight into the unpredictable but still deterministic be- havior of these systems. Characterization of strange at- tractors with fully developed chaotic dynamics has been done through computation of invariants—metric, dy- namical and topological. Metric invariants (Grassberger & Procaccia, 1983) involve statistical characterization of the distribution of points in phase space (via., evalua- tion of fractal dimensions of various kinds). Dynamical invariants like the Lyapunov exponents (Abarbanel, Brown, Sidorowich, & Tsimring, 1993) are based on the evolution properties of the trajectories and study invariant system properties under coordinate transformations. However, the determination of both types of invari- ants has limitations for varying reasons. This is mainly * Corresponding author. Tel.: +91-20-589-3095; fax: +91-20- 589-3041. E-mail address: bdk@ems.ncl.res.in (B. D. Kulkarni). because the analysis requires long and noiseless data and involves high computational eort. Further, dynamical invariants provide little information on the mechanisms governing these systems and they are not robust for vari- ation in the values of system control parameters. On the other hand, topological invariants are based on ‘stretching’ and ‘folding’ mechanisms present in the dynamics and has many advantages to oer besides com- plementing that gained by metric and dynamical invari- ant ananlyses (Boulant, Lefranc, Bielawski, & Derozier, 1997a, Gilmore, 1998). Topological invariants being sta- ble under parametric variations allow system features to be veried independently. Indeed, a global characteriza- tion of system dynamics may be obtained in the form of templates that allow dierent systems to be proven equivalent or otherwise. Also, bifurcation selection rules, for creation and annihilation of unstable periodic orbits (Lathrop & Kostelich, 1989), in chaotic dynamics can be studied. Topological analysis can also be used to vali- date models created by parameter estimation from chaotic time series data (Letellier et al., 1998). However, absence of topological invariants in dimensions greater than three 0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII:S0009-2509(01)00160-9