Hopf Bifurcations in a Nonlinear Two-Dimensional Autonomous System

Requires a Wolfram Notebook System

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

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

In the study of nonlinear dynamics, it is useful to first introduce students to simple systems that exhibit periodic behavior as a consequence of a Hopf bifurcation. The two-dimensional nonlinear and autonomous system given by



has this feature. These equations describe the autocatalytic reaction of two intermediate species and in an isothermal batch reactor, when the system is far from equilibrium [1]. In this context, the steady state referred to below is a pseudo steady state, and is applicable when the precursor reactant is slowly varying with time.

The unique steady state is given by and . This steady state is at the position of the green dot in the phase portrait diagram. It appears as the intersection of the dotted blue and green curves, which are the level curves given by and .

The stability of steady state to small disturbances can be assessed by determining the eigenvalues of the Jacobian . A Hopf bifurcation occurs when a complex conjugate pair of eigenvalues crosses the imaginary axis.

This Demonstration shows the phase portrait with the vector field of directions around the critical point . In addition, the eigenvalues of , the trace , the determinant , and are computed. The time series are also plotted in a separate diagram (the blue and red curves correspond to and , respectively. Finally, the real and imaginary parts of the two eigenvalues and are plotted (in blue and red, respectively) versus for user-specified values of the parameter .

To observe a Hopf bifurcation, you can set and vary . As you can see in the snapshots, there are two Hopf bifurcation points. Indeed, for , the steady state is a stable focus; for , the trajectory is attracted to a stable periodic solution (called a limit cycle); and finally for , the steady state is again a stable focus. For , the two Hopf bifurcation points are obtained at and . Finally, the last snapshot (obtained for larger values of , for instance, ), the Hopf bifurcation disappears and there are only transitions from a stable node to a stable focus and then back to stable node.


Contributed by: Housam Binous, Ahmed Bellagi, and Brian G. Higgins (October 2013)
Open content licensed under CC BY-NC-SA




[1] P. Gray and S. K. Scott, Chemical Oscillations and Instabilities: Non-Linear Chemical Kinetics, Oxford: Clarendon Press, 1994.

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.