Strange Attractors for Oberbeck–Boussinesq Model

Vakulenko, Sergey

doi:10.1007/s10884-020-09939-z

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…This property implies that semiflows can generate all structurally stable dynamics (up to orbital topological equivalency). Among the systems enjoying UDA, there are a number of fundamental ones: quasilinear parabolic equations 26 , 36 , time-continuous and time-recurrent neural networks 37 , a large class of reaction-diffusion systems with heterogeneous sources 9 , generalized Lotka-Volterra system 38 , Oberbeck-Boussinesq model 39 . Also, the Euler equations on multidimensional manifolds exhibit similar properties 32 .…”

Section: Methodsmentioning

confidence: 99%

Robust morphogenesis by chaotic dynamics

Reinitz

Vakulenko

Sudakow

et al. 2023

Sci Rep

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This research illustrates that complex dynamics of gene products enable the creation of any prescribed cellular differentiation patterns. These complex dynamics can take the form of chaotic, stochastic, or noisy chaotic dynamics. Based on this outcome and previous research, it is established that a generic open chemical reactor can generate an exceptionally large number of different cellular patterns. The mechanism of pattern generation is robust under perturbations and it is based on a combination of Turing’s machines, Turing instability and L. Wolpert’s gradients. These results can help us to explain the formidable adaptive capacities of biochemical systems.

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Section: Methodsmentioning

confidence: 99%

Robust morphogenesis by chaotic dynamics

Reinitz

Vakulenko

Sudakow

et al. 2023

Sci Rep

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“…In more flexible spaces than the Euclidean 3-dimensional one, let us mention the work [TdL21] of Torres de Lizaur who constructed an embedding of any smooth flow into the time-dependent evolution of the Euler PDE on high-dimensional Riemannian manifolds. See also [Vak21] for other embedding results in the context of the Oberbeck-Boussinesq model of the Navier-Stokes equations (on 2-dimensional domains) with external parameters (a space-dependent external force and heat source).…”

Section: Contributions To the State Of The Art On Complexity Of Stead...mentioning

confidence: 99%

Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics

Berger¹,

Florio²,

Peralta‐Salas³

2022

Preprint

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Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set G of stationary solutions to the Euler equations in R 3 such that each vector field X ∈ G is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of X at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension 3 or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.

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“…The main idea to simplify the formula (31) for G i is as follows. Suppose that the kinks oscillate at certain fixed points Xj , i.e., X i (t) = Xi + Xi (t), (32) where Xi are new unknowns.…”

Section: Hopfield Systemmentioning

confidence: 99%

“…Such approach was used for finite dimensional systems (see [28]), but it can be extended on infinite dimensional evolution equations. This RVF approach is developed by first P. Poláčik to prove existence of non-trivial large time behaviour for quasilinear parabolic equations (see [29,30]) and developed in [12,24] for reaction-diffusion systems, in [18] for neural networks and in [31] for weakly compressible Navier-Stokes equations.…”

Section: Appendix Realisation Of Vector Fields (Rvf)mentioning

confidence: 99%

Chaotic waves serve as universal pattern generators

Vakulenko,

Sudakow,

Reinitz

et al. 2021

Preprint

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Strange Attractors for Oberbeck–Boussinesq Model

Cited by 6 publications

References 20 publications

Robust morphogenesis by chaotic dynamics

Robust morphogenesis by chaotic dynamics

Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics

Chaotic waves serve as universal pattern generators

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