# Tag: Time Series

• ## Multidimensional Scaling

Multidimensional Scaling (MDS) is a linear embedding method used when we only know the pairwise distances between data points. For linear systems, MDS works well with as little as 10 points and the system is 2 dimensional.

• ## Estimating Lyapunov Spectra of ODEs using Python

Python code is shown that estimates the Lyapunov spectra for the Rossler and Lorenz systems. This code is written in a way that makes it adaptable for other continuous-time systems.

• ## Modelling Chaotic Systems using Support Vector Regression and Python

Support Vector Regression is a technique in machine learning that can be used to model chaotic data. A program is shown to work on Delayed Henon Map data.

• ## Lyapunov spectra of inverted discrete dynamical systems

One can estimate the lyapunov spectrum of dynamical systems and their inverted counterparts using local Jacobian matrices and Wolf’s algorithm.

• ## Generating Poincaré Sections and Return Maps

Using the iterative 4-th order Runge-Kutta method as described here, we can create low dimensional slices of the system’s attractor known as Poincare Sections, Return maps, or Recurrence maps. We will use the Rossler attractor for this example, with a, b, and c set to 0.2, 0.2, and 5.7, respectively. Poincare sections are important for visualizing…

• ## Generating time series for Ordinary Differential Equations

We can produce a time series from ordinary differential equations by solving the equations using the iterative 4-th order Runge-Kutta method and plugging each of the solutions back into the equations.

• ## Modelling Sensitivity using Neural Networks

Artificial neural networks can be applied to the delayed Henon map and shown to replicate the sensitivities of the map surprisingly well.

• ## Inverted Delayed Henon Map

Inverting the delayed Henon map yields a repellor whose sensitivities can be explored.

• ## Delayed Henon Map Sensitivities

Partial derivatives can be used to explore how sensitive the output of a function is to perturbations in each of the time lags.