Posts

Lorenz Attractor We start with exploring Lorenz differential equations (also known as Lorenz attractor) using Python through examples, look at extremely complicated, aperiodic, non-transient and locally unstable trajectories created by Lorenz attractor. Read more...

Poincaré Surface of Section Poincaré surface of section (also referred to as Poincaré section or Poincaré map) is powerful technique extracting fundamental properties from flows in the form of a discrete maps. A trajectory or set of trajectories are sampled periodically, by looking at successive intersections with a plane in the phase space. Read more...

Fractal Dimension In the study of dynamical systems there are many quantities that identify as "entropy". Notice that these quantities are not the more commonly known thermodynamic ones, used in Statistical Physics. Rather, they are more like the to the entropies of information theory, which represents information contained within a dataset, or information about the dimensional scaling of a dataset. Based on the definition of the generalized entropy, one can calculate an appropriate dimension. Read more...

Lyapunov Exponents Lyapunov exponents measure exponential rates of separation of nearby trajectories in the flow of a dynamical system. Read more...

Delay Embedding We try to reconstruct a chaotic dynamical system from a time-series using Taken's delay embedding technique. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms). Read more...

Recurrence Plots A recurrence plot is a way to quantify recurrences that occur in a trajectory. What's a recurrence? As the name suggests, a recurrence happens when a trajectory visits the same neighborhood on the phase space that it was at some previous time. The recurrence plot is a visual representation of a sparse square matrix of boolean values (called recurrence matrix). Read more...