Two-dimensional Phase Portrait

Interactive visualization of two-dimensional dynamical systems

System Equations

Parameters

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View Range

Add Trajectory

Some information

  • Use "dot", not "comma" for decimals.
  • Constants: Euler's number is e and pi is PI.
  • Basic Functions: abs(x), sqrt(x), exp(x), log(x) (natural log), and pow(x, y).
  • Trigonometry (Radians): sin(x), cos(x), tan(x) and their inverses arcsin(x), arccos(x), arctan(x), arctan2(y, x).
  • Hyperbolic: sinh(x), cosh(x), tanh(x).
  • If the trajectory doesn't seem right, try to switch over to 4th order Runge-Kutta integration by pressing the "RK4" button. Trajectories generated using Euler integration might generate lines that are not correct.
  • When pressing somewhere on the plot/graph: a window will pop up underneath it where you automatically get the (numerically) calculated eigenvalues for that point. Assuming it was a fixed point that was clicked, the stability analysis should give a hint of what type of a fixed point it is. Use nullclines to find fixed points, it should be where they are crossing each other (fixed points occur where both nullclines intersect).
  • Long integration time and/or many trajectories can be computationally expensive, be cautious if you are on a mobile or on a low performance device.

2D Matrix eigenvalues and eigenvectors

After you have found the fixed points (analytically), you can insert the stability matrix (Jacobian matrix) here and get the eigenvalues and eigenvectors for that fixed point.

Stability Analysis (Jacobian)

Why I made this? Well, I had a course in dynamical systems and I passed the course examination with not so good phase-portraits and since it is a recurring concept in various fields, I made this webpage where you can just insert a 2 dimensional system and play around with it visually. It's worth reminding you that this is purely numerical and analytical solutions are always more accurate and you still have to do some analytical work or use symbolic algebra system to get the bifurcation diagrams needed to understand what is going on with the parameters and their relationships. I chose 5 parameters since if you have more than 5, well... you should probably consider something more powerful than a simple webpage. Books used when I did the course:

  • Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (3rd edition) by Steven H Strogatz | ISBN: 9780367026509

  • Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes | ISBN: 9781461270201

  • Chaos: Classical and Quantum (Edition 17) Predrag Cvitanovic, Roberto Artuso, Ronnie Mainieri, Gregor Tanner and Gábor Vattay | July 19, 2020

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