Data-Driven-Dynamical-Systems

Dynamical systems numerical analysis, DMD and Koopman Operator theory

Dynamical Systems:

$$ \frac{d}{dt}\mathbf{X}(t) = \mathbf{f}(x(t),t;\beta) $$

where $x$ is the state of the system, f is the dynamics (vector fields), t is time and $\beta$ is the parameters.

Example : Consider Lorentz system of equations:

$$ \begin{aligned} \dot{x}&=\sigma(y-x)\\ \dot{y}&= x(\rho-z)-y\\ \dot{z}&= y - \beta z, \end{aligned} $$

The system is simulated by simu_lorenz.m code. Where the parameters are set as $\sigma = 20$, $\rho = 28$ and $\beta = 8/3$. Time step is set as $\delta t = 0.004$ from $t = 0$ to $t = 60$ and initial conditions are $x(0) = 0$, $y(0) = 1$ and $z(0) = 20$.

Discrete-time systems

Consider the Discrete-time dynamic systems:

$$ \mathbf{X}{k+1} = \mathbf{F}(X{k}) $$

Example: Logistic equation

$$ \mathbf{X}{k+1} = \beta \mathbf{X}{k}(1 - \mathbf{X}_{k}) $$

We can simulate the attractors using logistic_attractors.m based on the varrying $\beta =0:4$ as shown in the graph below: