Trace-Determinant Plane

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The stability of solutions of a linear system of differential equations with constant coefficients can be classified using the trace and determinant of the coefficient matrix, without having to solve the system. Here the system in question is of the form , where , , and are real parameters. The stability of solutions of this system exhibit a bifurcation at , and solutions are qualitatively different depending on whether , , or . This Demonstration represents the stability of solutions in the -plane as varies. Drag the locator to identify the different regions.

Contributed by: John Elliott (July 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Chapter 4 of [1] provides a readable account of these ideas and was the inspiration for this Demonstration.

Reference

[1] M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, San Diego: Elsevier, 2004.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send