Abstract
Dissipative dynamical systems often possess multiple coexisting attractors. The set of initial conditions leading to trajectories landing on an attractor is the basin of attraction of this attractor. Each attractor thus has its own basin, which is invariant under the dynamics, since images of every point in the basin still belong to the same basin. The basins of attraction are separated by boundaries. We shall demonstrate that it is common for nonlinear systems to have fractal basin boundaries, the dynamical reason for which is nothing but transient chaos on the boundaries. In fact, fractal basin boundaries contain one or several nonattracting chaotic sets.
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Notes
- 1.
Near O, we have \(V \approx -{s}^{2}{x}^{2}/2\). The solution to (5.1) is \(x(t) = {c}_{+}{\mathrm{e}}^{{\lambda }_{+}t} + {c}_{-}{\mathrm{e}}^{{\lambda }_{-}t}\) with \({\lambda }_{\pm } = -\gamma /2 \pm { ({s}^{2} + {\gamma }^{2}/4)}^{1/2}\). Thus, for \({c}_{+} = 0\), we have \(v(t) = {\lambda }_{-}x(t) \sim {\mathrm{e}}^{{\lambda }_{-}t}\) and x(t) → 0, v(t) → 0 as t → ∞, along the line \(v = {\lambda }_{-}x\)
- 2.
Choose a phase-space region R of nonzero volume that encloses an attractor. That the system is dissipative means that the inverse dynamics is volume-expanding. Since R is completely in the basin of attraction, all its preimages are in the basin as well. In the limit t → − ∞, the volume of the preimage becomes infinite.
- 3.
Consider an open neighborhood \(\mathcal{B}\) of one of the attractors at infinity. Choose a point p in its basin and evolve it forward in time. Eventually, the resulting trajectory will approach the attractor, which means that at some finite time, the trajectory will enter \(\mathcal{B}\), say at point p′. The point p′ in \(\mathcal{B}\) must then have an open neighborhood. Since p′ is iterated from p in finite time, p must also have an open neighborhood in the basin.
- 4.
Since very close to a boundary arises the chaotic saddles’s stable manifold is nearly space-filling, the set of initial conditions leading to long transients also exhibits riddled-like behavior [834]
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© 2011 Springer Science+Business Media, LLC
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Lai, YC., Tél, T. (2011). Fractal Basin Boundaries. In: Transient Chaos. Applied Mathematical Sciences, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6987-3_5
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DOI: https://doi.org/10.1007/978-1-4419-6987-3_5
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