Abstract
The Hopf bifurcation theorem is proved in the roundabout way adopted in Hassard, Kazarinoff and Wan’s 1981 book, proceeding by way of normal form theory and the centre manifold theorem before the final proof using the implicit function theorem. There is then a thorough discussion of secondary Hopf bifurcation, leading to the ideas of Arnold tongues and frequency locking, and the concept of tertiary Hopf bifurcation is briefly raised and dismissed.
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Notes
- 1.
Well, not the simplest; but it’s the simplest interesting bifurcation.
- 2.
The notation \(\phi _{s} \circ \phi _{t}(x)\) means \(\phi _s[\phi _t(x)]\).
- 3.
It is a periodic orbit with infinite period.
- 4.
By this, we mean that there is a single eigenvector corresponding to the passage of the eigenvalue through zero; by making a suitable coordinate change we can make this eigenvector one of the coordinate axes, and we call the corresponding coordinate an eigencoordinate.
- 5.
- 6.
If the unperturbed \((\varepsilon = 0)\) system is itself nonlinear, then the first of these is nonlinear, and it is usually appropriate to define the three variables in a more complicated way: this is Kuzmak’s method.
- 7.
That is, it has derivatives of all orders. One might think this is the same as being analytic, i.e. having a convergent Taylor series, but it is not the case: think of \(\exp (-1/x^2)\) at the origin.
- 8.
The inner product is just the generalisation of scalar product in vector spaces to more complicated spaces, such as function spaces; here, as we are in \(\mathbf{R}^n\) (actually \(\mathbf{R}^2\) in this example), they are the same.
- 9.
For a vector \(x=(x_1,\ldots , x_n)\in \mathbf{R}^n\), we use a notation \(O(x^k)\) to represent \(O(x_1^{\alpha _1}\ldots x_n^{\alpha _n})\), \(\sum _i\alpha _i=k\), and we also write \(O(x^k)=O(k)\).
- 10.
A need not, in fact, be diagonisable.
- 11.
A graph is the generalisation to n dimensions of the two-dimensional idea of the graph of a scalar function h(x) of a scalar variable x.
- 12.
Transverse here has the obvious geometrical meaning. If U intersects \(u_{\!p}\) at \(u=u^{*}\), and \(\hat{u}\) is a vector tangent to \(u_{\!p}\) at \(u^{*}\), then we require \(\langle u, v\rangle \) to be non-zero for non-zero vectors v in U (using the usual inner product).
- 13.
In fact, the centre manifold theorem applies to maps as well, and so one can prove that bifurcations associated with passage of \(\lambda \) through \(\pm 1\) are strictly analogous to the one-dimensional maps discussed in Chap. 2.
- 14.
The symbol \({\mathop {=}\limits ^{\triangle }}\) indicates definition.
- 15.
The symbol \({\mathop {=}\limits ^{\triangle }}\) indicates definition, as earlier in (3.111).
- 16.
Note that the terms in (3.138) are nonlinear, so that \(m_1+m_2\ge 2\).
- 17.
The resonance referred to here is that between primary and secondary periods, not that involved in removing nonlinear terms from the system.
- 18.
A function of time is doubly periodic if it can be written in the form \(f(\omega _1t,\omega _2t)\), where f(x, y) is \(2\pi \)-periodic separately in each of its arguments, and \(\omega _1/\omega _2\) is irrational.
- 19.
Using the continued fraction approximation for an irrational number.
- 20.
It will be less than this if p/q is near either end point.
- 21.
At this point, we are practically at KAM theory. We will come back to this in Chap. 5.
- 22.
As mentioned in Chap. 1, a set U is dense in X if X is the closure of U.
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Fowler, A., McGuinness, M. (2019). HopfBifurcations . In: Chaos. Springer, Cham. https://doi.org/10.1007/978-3-030-32538-1_3
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DOI: https://doi.org/10.1007/978-3-030-32538-1_3
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