Abstract
In this chapter, we formulate conditions defining the simplest bifurcations of equilibria in n-dimensional continuous-time systems: the fold and the Hopf bifurcations. Then we study these bifurcations in the lowest possible dimensions: the fold bifurcation for scalar systems and the Hopf bifurcation for planar systems. Appendixes A and B are devoted to technical questions appearing in the analysis of Hopf bifurcation: Effects of higher-order terms and a general theory of Poincaré normal forms, respectively. Chapter 5 shows how to “lift” the results of this chapter to n-dimensional situations.
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Notes
- 1.
The vectors \(q(\alpha )\) and \(p(\alpha )\), corresponding to the simple eigenvalues, can be selected to depend on \(\alpha \) as smooth as \(A(\alpha )\).
- 2.
Actually, only the “resonant” cubic term of the inverse is required
$$ w=z - \frac{h_{20}}{2}z^{2}-h_{11}z \bar{z}-\frac{h_{02}}{2} \bar{z}^{2} + \frac{1}{2}(3h_{11}h_{20}+2|h_{11}|^{2}+|h_{02}|^{2})z^{2}\bar{z} + \cdots , $$where the dots now mean all undisplayed terms.
- 3.
- 4.
It is always useful to express the Jacobian matrix using \(\omega \) since this simplifies expressions for the eigenvectors.
- 5.
Another way to compute \(g_{20},g_{11}\), and \(g_{21}\) (which may be simpler if we use a symbolic manipulation software) is to define the complex-valued function
$$ G(z,w)=\bar{p}_1F_1(zq_1+w\bar{q}_1,zq_2+w\bar{q}_2)+\bar{p}_2F_2(zq_1+w\bar{q}_1,zq_2+w\bar{q}_2), $$where p, q are given above, and to evaluate its formal partial derivatives with respect to z, w at \(z=w=0\), obtaining \( g_{20}=G_{zz},\ g_{11}=G_{zw}\), and \(g_{21}=G_{zzw}\). In this way no multilinear functions are necessary. See Exercise 4.
- 6.
Some implementations of MAPLE may produce the eigenvectors in a different form.
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Kuznetsov, Y.A. (2023). One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems. In: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-031-22007-4_3
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DOI: https://doi.org/10.1007/978-3-031-22007-4_3
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