How to create phase portraits ?

Introduction to Phase Portraits

A phase portrait offers a visual representation of the trajectories of a dynamical system within a two-dimensional phase space. Observing these trajectories allows for the inference of the system's dynamics, such as stability, chaos, or periodicity.

Setting Up a Phase Portrait

First step is to create a model in jinkō . Then in the model view, you can select "Phase portrait" tab and fill in those inputs :

• Freeze model toggle: When set to true, only the two selected left-hand side (LHS) variables will be solved, and the rest of the model will remain constant (a.k.a. frozen). When set to false, the rest of the model will be solved jointly, which may increase computation time depending on the model size.
• Left-hand side (x-axis), Left-hand side (y-axis): These are the variables of the model observable in the phase space.
• Min and Max: This defines the range of initial values to explore for each variable. Trajectories will be generated starting from initial conditions within these ranges.
• Grid points per dimension: This determines the division of the phase space along each axis. A higher number indicates more initial conditions and, consequently, more trajectories.

Analysis Methodology

The primary graphical output displays the system's trajectories over time from various initial conditions. 

Tips on how to use the feature:

• It is recommended to observe patterns such as fixed points, limit cycles, or chaotic attractors. The arrangement and shape of these trajectories can offer insights into the system's dynamics.
• It is suggested to adjust various ranges and grid sizes to assess their impact on the phase portrait.
• Analyzing the system's sensitivity to initial conditions is important by noting how closely spaced trajectories diverge.
• Utilizing the phase portrait to develop hypotheses about the nature of the dynamical system is beneficial. These hypotheses can then be tested through further simulations.

Example

Displayed is a phase portrait for the Lorenz Model over a simulation period of 30 seconds, set up with three different initial conditions.

The plot demonstrates a butterfly-shaped pattern for both initial conditions, indicative of chaotic behavior.