2020
DOI: 10.1137/18m1235818
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Abstract: Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modelling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte-Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets… Show more

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Cited by 6 publications
(9 citation statements)
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References 38 publications
(86 reference statements)
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“…Moreover, the definition of Z 1 together with (7a) imply that a * x is invertible, therefore so is a (because 1 η is a commutative ring), and we do have that A in injective. We can thus apply Theorem 2.14, and according to (8), there exists a unique zero x * of F in B 1 η (x, r) for all r ∈ [r min , r max ), with r min = 0.073908425226395, r max = 0.781015206412929.…”
Section: A Basic Examplementioning
confidence: 99%
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“…Moreover, the definition of Z 1 together with (7a) imply that a * x is invertible, therefore so is a (because 1 η is a commutative ring), and we do have that A in injective. We can thus apply Theorem 2.14, and according to (8), there exists a unique zero x * of F in B 1 η (x, r) for all r ∈ [r min , r max ), with r min = 0.073908425226395, r max = 0.781015206412929.…”
Section: A Basic Examplementioning
confidence: 99%
“…Most of the material presented in this subsection about the relationships between orthogonal polynomials and Banach algebras can be found (in a different context) in the lecture notes [42]. We also refer to the appendix of [8] for discussions related to implementation issues.…”
Section: Linearization Formulas and Generalized Convolution Productsmentioning
confidence: 99%
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“…Only very recently, the link between parametric uncertainty and bifurcations has emerged as a sub-area within the theory of nonlinear dynamics [2,12,15,16]. This link is important since parametric uncertainty leads to uncertainty in the location of tipping points and the shape of the bifurcation curve.…”
Section: Introductionmentioning
confidence: 99%
“…Recently, substantial interest has been focused on understanding the interplay between stochasticity and bifurcations. Yet, the setting in almost all of these works is focused on either stochastic differential equations (SDEs) involving (space-)time stochastic forcing processes [2,13], or less frequently on random differential equations (RDEs) with a fixed random parameter distribution [14,42]. Particularly interesting dynamics seems to appear for SDEs in oscillatory situations [6,20,46].…”
Section: Introductionmentioning
confidence: 99%