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As it has been mentioned, this transition happens at the Bogdanov-Takens points. Thus, there should exist an unstable limit cycle, bifurcating from the equilibrium. Activate the parameter k7. The Hopf bifurcation curve: Two-parameter bifurcation analysis of equilibria and limit cycles with matcont Yu.

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In the Numeric window, you can read the corresponding values of k7k8. You should get the central S-shaped curve presented in Figure 1.

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Stop computations and rename the computed curve into Equilibrium. Input the following numerical data in the appearing Starter window: Find the period-doubling PD of the cyle bifurcating from the Hopf point and continue it in two parameters.

Resume and extend until the window is left. Consider the following prey-predator system from mathematical ecology by Bazykin and Khibnik []: Rename this curve in Hopf.

The edge between two fold curves is a parameter region where the system has three equilibria. In a narrow interval of Q2-values between two LP-points the system has three equilibria.

Resume the countinuation and terminate it when the Hopf curve leaves the window.

Change the plotting region of the 2Dplot window to Abscissa Ordinate 1 0. Compute the two-parameter bifurcation diagram in the m, n -plane and sketch all qualitatively di?

Then choose Type Curve Hopf to prepare for the continuation of the Hopf curve. The computed curve will also contains two extra codim 2 points labeled by R1.

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The Hopf bifurcation curve: Close the Curves window. The equilibrium curve will be continued and you get a Hopf bifurcation. There are two more codim 2 bifurcation points at the Hopf point: Hopf point in the curve Equilibrium - as initial.

Starter window for the continuation of the limit point curve. In the Continuer window, decrease the maximal step size along mmatcont curve to MaxStepsize 0.

The new Starter window for the continuation of a Hopf jatcont will appear automatically, see Figure 4.

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In the Continuer window, set MaxStepsize equal to 2. Consider the adaptive control system of? Click Compute Forward and wait until the computed NS-curve enters the window this takes some time. At the fold-Hopf both the trace matcontt the determinant of the Jacobian matrix of 1 vanish. Each point in the curve is a limit point for the equilibrium curve at the corresponding value of K.

Generate the derivatives of order 1,2, and 3 symbolically. Hopf and limit point of cycles curves connecting the GH-points.

Limit point in the? These are strong resonance 1: Activate one more parameter, namely k8, in the Starter window.

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There is now a big interval of the parameter values, in which there are three equilibria in the system. Consider the following chemical? The ratio We will matfont matcont to continue equilibria and their bifurcations in the following chemical model that describes CO oxidation on platinum: Derive an analytic expresssion for the Hopf bifurcation curve in 3.