Abstract
Homoclinic bifurcation theory is discussed, beginning with the Lorenz equations as an example, and the corresponding two-dimensional Poincaré map and its one-dimensional approximation are derived. Symbolic dynamics and the Smale horseshoe are described, and then Shilnikov bifurcations for saddle-focus homoclinic bifurcations are analysed. The final section generalises the results to n-dimensional flows, and mentions the corresponding results for infinite-dimensional (partial differential equation) flows, without detail. The possible relation to fluid turbulence is mentioned.
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Notes
- 1.
We can in fact assume that these points are exact by making a further change of variable, so that the stable manifold is locally given by the \((x_{2}, x_{3})\) plane, and the unstable manifold is the \(x_{1}\) axis. The procedure is similar to the centre manifold construction. But if the box is small (\(|c_{i}| \ll 1)\), the error is small anyway, and the formal construction involved is identical.
- 2.
An affine map is simply a linear, inhomogeneous map.
- 3.
In fact, this can be formally derived by simply linearising the flow map from \(S'_\pm \) to S.
- 4.
A Cantor set in an interval is one which is closed and uncountable but has no interior points and no isolated points.
- 5.
Essentially, this means that points in the set can be written as a coordinate pair \((x_u, x_s)\), with \(x_u\in M_u\) and \(x_s\in M_s\).
- 6.
Because we are solving ordinary differential equations to produce the Poincaré map; if we are solving partial differential equations, we can still manage but it requires some artistry.
- 7.
The orbit approaches every point in I arbitrarily closely.
- 8.
Fundamental matrices were in effect defined for linear systems of ordinary differential equations at (3.114).
- 9.
The one \(\propto te^{tD_0}\).
- 10.
This effectively places v on the (outgoing) face S of the box near 0, although it may be simply thought of as a convenient transformation.
- 11.
Remember that D is diagonal.
- 12.
In this context, the co-dimension refers to the number of independent parameters whose selection is necessary to secure the bifurcation.
- 13.
See Sect. 4.5.
- 14.
A finite domain can be effectively infinite if the smallest relevant dynamical length scale is much smaller than the domain size; this commonly occurs when the dynamical forcing parameter (e.g. the Reynolds number in shear flows, or the Rayleigh number in convection) is large.
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Fowler, A., McGuinness, M. (2019). Homoclinic Bifurcations. In: Chaos. Springer, Cham. https://doi.org/10.1007/978-3-030-32538-1_4
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DOI: https://doi.org/10.1007/978-3-030-32538-1_4
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