Transformation Invariance of Lyapunov Exponents

Lyapunov exponents represent important quantities to characterize the properties of dynamical systems. We show that the Lyapunov exponents of two di€erent dynamical systems that can be converted to each other by a transformation of variables are identical. Moreover, we derive sucient conditions on the transformation for this invariance property to hold. In particular, it turns out that the transformation need not necessarily be globally invertible. Ordinary di€erential equations constitute a widely used tool to describe the dynamical behavior of physical, biological, chemical and many other systems. Moreover, since Lorenz's [1] discovery of deter-ministic non-periodic ¯ow, time-continuous dynamical systems play an important role in the exploration of chaotic phenomena [2±5]. In the modern theory of dynamical systems, their properties are mainly analyzed in a qualitative way in terms of their ¯ow in phase space. A ®rst simpli®cation of such analysis might be achieved by transforming the investigated dynamical system to a system with a functional simpler or more convenient form. As a speci®c example, we mention recent advances [6,7] in the theory of three-dimensional dynamical systems with quadratic non-linearities where coordinate transformations to the so-called jerky dynamics allow for a classi®cation based on functional simplicity of the resulting third-order di€erential equations. The new dynamical system, however, should have the same dynamical properties as the original one, i.e., the character of the long-time dynamics (®xed point, limit cycle, strange attractor etc.) should not be changed. This leads to the question about the invariance properties of such quantities that can be used to characterize di€erent dynamical long-time behavior, such as dimensions of attractors or Lyapunov exponents. The concept of the dimension of an attractor is based on its metric properties, leading, e.g., to the Hausdor€ dimension, or on its invariant measure, yielding, e.g., the information dimension [8]. With this concept a simple classi®cation of attractors is possible. For instance, a ®xed point has dimension zero, a stable limit cycle has dimension one, a 2-torus has dimension two, while the dimensions of strange at-tractors being a signature of dissipative, deterministic chaos take on values that are typically non-integer. In Ref. [9], it is shown that the Hausdor€ dimension as well as the information dimension are invariant under a wide class of invertible coordinate transformations. Lyapunov exponents are de®ned using the dynamical long-time properties of the trajectories on an at-tractor [10]. The type of the attractor is uniquely characterized by its Lyapunov spectrum, i.e., the …