Abstract
In this chapter, we treat transiently chaotic dynamical systems under the influence of noise, focusing on a number of physical phenomena. Firstly, we will demonstrate that noise can increase the lifetime of transient chaos and induce dynamical interactions among different invariant sets of the system. As a result, the stationary distributions of dynamical variables in a noisy system can be much more extended in the phase space than those in the corresponding deterministic system. Secondly, if the system has a nonchaotic (e.g., periodic) attractor but there is transient chaos due to a coexisting nonattracting chaotic set, noise can cause a trajectory to visit both the original attractor and the chaotic saddle, leading to an extended chaotic attractor. This is the phenomenon of noise-induced chaos, which can arise, for instance, when the dynamical system is in a periodic window. Of particular interest is how the Lyapunov exponent and other ergodic averages scale with the noise strength. Thirdly, if the system has a chaotic attractor, noise can cause trajectories on the attractor to move out of its basin of attraction so that either the attractor is enlarged or the originally attracting motion becomes transient. This is the phenomenon of noise-induced crisis, dynamically due to noise-induced heteroclinic or homoclinic tangencies that cause the attractor to collide with its own basin boundary. An issue of both theoretical and experimental interest is how the average transient lifetime depends on the noise strength.
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- 1.
Although in some experiments [70] and in time-delayed systems [850] no apparent effect of noise on transient chaos has been found.
- 2.
For noise of order r, \(W(\mathbf{x}) \sim Z(\mathbf{x}){\mathrm{e}}^{-\Phi (\mathbf{x})/{\sigma }^{r} }\).
- 3.
For noise of order r, the first equation should be replaced by \({\mathbf{x}}_{n+1} = \mathbf{f}({\mathbf{x}}_{n},p) + {\eta }_{n+1}^{1/(r-1)}\), for r even.
- 4.
For noise of order r, \(\Phi (\mathbf{x}) =\min { \sum \nolimits }_{n=0}^{\infty }{\frac{1} {r}{\left ({\mathbf{x}}_{n+1} -\mathbf{f}({\mathbf{x}}_{n},p)\right )}^{r}}_{\mid {\mathbf{x}}_{\infty }\equiv \mathbf{x}} + \mbox{ constant}\).
- 5.
In the case of fractal basin boundaries, the exit point is a point of the nonattracting chaotic set belonging to the fractal boundary.
- 6.
For noise of order r, \(k \sim {\mathrm{e}}^{-\Delta \Phi /{\sigma }^{r} }\).
- 7.
For fractal boundaries the exit point x e is on the quasipotential plateau, and hence ΔΦ is the difference between its values on the plateau and on the attractor.
- 8.
The role of a chaotic saddle in enhancing the exit rate suggests that when a dynamical system undergoes a basin boundary metamorphosis (see Sect. 5.4.1) by which a smooth boundary becomes fractal so that a nonattracting chaotic set arises on the boundary, the rate of exiting the basin due to noise can be enhanced significantly. This has indeed been observed [729].
- 9.
For noise of order r, we have \({\sigma }_{\mathrm{c}} = {[\Delta \Phi /\ln (Z/\chi )]}^{1/r}\).
- 10.
For noise of order r, \(\Delta \Phi = c{(p - {p}_{\mathrm{b}})}^{r-1/2}\), and \({t}_{\mathrm{e}}(p) \sim {(p - {p}_{\mathrm{b}})}^{-1/2}g[{(p - {p}_{\mathrm{b}})}^{1-1/(2r)}/\sigma ]\) with g(z) ∼ exp(cz r).
- 11.
The Lyapunov exponents are the time-averaged stretching or contracting rates of infinitesimal vectors along a typical trajectory in the phase space, which can be defined for both deterministic and stochastic dynamical systems.
- 12.
The scaling law and the exponent remain unchanged even if the threshold is not neglected [776].
- 13.
The noise strength σc corresponds to (4.21) with \(\chi /Z = \kappa {\tau }_{0}\mid {\lambda }_{1}^{P}\mid /{\lambda }_{1}^{C}\).
- 14.
This consideration does not apply to nonautonomous systems, for which there is always a neutral direction along the time axis and therefore always a zero Lyapunov exponent.
- 15.
For noise of order r, we have ΔΦ(Δx) ∼ Δx r and \(\tau \sim {\sigma }^{-\gamma }\exp \left (\frac{c{({p}_{\mathrm{c}}-p)}^{r}} {{\sigma }^{r}} \right )\).
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© 2011 Springer Science+Business Media, LLC
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Lai, YC., Tél, T. (2011). Noise and Transient Chaos. In: Transient Chaos. Applied Mathematical Sciences, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6987-3_4
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DOI: https://doi.org/10.1007/978-1-4419-6987-3_4
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