Lifetimes of noisy repellors

Faisst, Holger; Eckhardt, Bruno

doi:10.1103/physreve.68.026215

Cited by 11 publications

(10 citation statements)

References 15 publications

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“…Using (15) and (16) we explicitly show that the decay rate of the first return time distribution of a set E in a closed system, given by the leading pole of F (z), is equal to the escape rate of the system open on E (given by the leading zero of ζ −1 op (z)), a previously heuristically derived result [1]. Furthermore, from Kac's lemma F ′ (1) = 1/µ B and thus, using (15) in (16) we have…”

Section: Small Hole Asymptoticsmentioning

confidence: 79%

“…On the contrary, in [9] it was observed that varying the position of the hole has a strong effect on the average lifetime of chaotic transients due to the complex periodic orbit structure in chaotic maps. This dependence on the position of the hole has generated a renovated interest among both physicists and mathematicians; in [10] the escape rate was shown to have non-trivial dependence on the position and non-monotonic dependence on the size of the holes, general results on the asymptotic behavior for small holes have been studied in [11,12,13,14] whilst in [15,16], the addition of noise is investigated. Open problems and reviews of this material can be found in [17,8,1].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Follow the fugitive: an application of the method of images to open systems

Cristadoro¹,

Knight²,

Esposti³

2013

J. Phys. A: Math. Theor.

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Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size-and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in general dynamical systems.

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Section: Small Hole Asymptoticsmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Follow the fugitive: an application of the method of images to open systems

Cristadoro¹,

Knight²,

Esposti³

2013

J. Phys. A: Math. Theor.

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“…An important task for the future will be to catalog and classify the many possible forms of interaction between nonlinearity and stochasticity, including noise‐induced chaos. For now, the list is still growing – we note in particular recent analyses of how noise can extend the lifetime of chaotic transients (Faisst and Eckhardt 2003) and of how chaotic saddles facilitate noise‐driven jumps between attractors (Kraut and Feudel 2002, 2003) – and ecologists are still largely unaware of it.…”

Section: Discussionmentioning

confidence: 99%

“…This occurs in empirically grounded models for host–pathogen (Rand and Wilson 1991, Billings and Schwartz 2002) and predator–prey dynamics (below). Moreover, noise can actually increase the typical length of time that trajectories remain near a chaotic repeller (Faisst and Eckhardt 2003). Chaotic repellers often become chaotic attractors at nearby parameter values, which is true in our case (Fig.…”

Section: What Is Noise‐induced Chaos?mentioning

confidence: 99%

When can noise induce chaos and why does it matter: a critique

Ellner¹,

Turchin

2005

Oikos

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Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ0 that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.

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“…Transient behaviour can be relatively persistent and can remain even after far exceeding the bifurcation value k c [55]. In addition the steady state can be sensitive to any perturbations which could cause the system to re-enter a potentially long chaotic transient again [56]. The behaviour of the two species system for k 11 ≥ k c seems to exhibit such chaotic transient behaviour with a quasi-periodic steady state (FIG.…”

Section: Dimensionality and Chaosmentioning

confidence: 99%

Evolutionary fields can explain patterns of high-dimensional complexity in ecology

2017

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One of the properties that make ecological systems so unique is the range of complex behavioural patterns that can be exhibited by even the simplest communities with only a few species. Much of this complexity is commonly attributed to stochastic factors which have very high-degrees of freedom. Orthodox study of the evolution of these simple networks has generally been limited in its ability to explain complexity, since it restricts evolutionary adaptation to an inertia-free process with few degrees of freedom in which only gradual, moderately complex behaviours are possible. We propose a model inspired by particle mediated field phenomena in classical physics in combination with fundamental concepts in adaptation, that suggests that small but high-dimensional chaotic dynamics near to the adaptive trait optimum could help explain complex properties shared by most ecological datasets, such as aperiodicity and pink, fractal noise spectra. By examining a simple predator-prey model and appealing to real ecological data, we show that this type of complexity could be easily confused for or confounded by stochasticity, especially when spurred on or amplified by stochastic factors that share variational and spectral properties with the underlying dynamics.

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Lifetimes of noisy repellors

Cited by 11 publications

References 15 publications

Follow the fugitive: an application of the method of images to open systems

Follow the fugitive: an application of the method of images to open systems

When can noise induce chaos and why does it matter: a critique

Evolutionary fields can explain patterns of high-dimensional complexity in ecology

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