Chaos and Noise in Dynamical Systems with Applications to Measurement

One of the best quantifiers of chaos for a dynamical system is the maximal Lyapunov exponent (denoted as LE), which measures the exponential rate at which nearby trajectories in phase space diverge. However, when noise is introduced into the system, it becomes unclear as to how the LE is even defined, or whether it is defined at all. For instance, when considering two nearby trajectories in phase space with noise, there is an inherent ambiguity as to whether the same noise realization should be taken for both trajectories when calculating the rate at which they exponentially diverge, or whether each trajectory should have its own noise realization. The first part of this thesis extends the definition of the LE to systems with noise, and shows that the treatment is dependent on whether the noise is internal or external. In particular, we show that the LE is still a well-defined notion in systems with noise, in that different noise realizations of the same noise do not affect the rate at which nearby trajectories diverge.

In the second part of the thesis, we will outline how to apply the techniques in Part I to study systems subjected to measurement. Because the term in the Lagrangian of a measured system can be treated as Gaussian white noise, the machinery developed in the first part will play a crucial role. We will explicitly calculate how the new measurement terms in the master equation affect the dynamics of the system, thus developing an intuition for what physical quantities the additional terms correspond to. We will finally describe how we plan to study the evolution of chaos in classical and quantum systems, which will shed some light on the quantum-classical transition of chaos. Because this is an ongoing project, the second part will mainly emphasize on how to use the results derived in the first part in future work.