"Large" strange attractors in the unfolding of a heteroclinic attractor

<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.</p>

The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms

The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

We show that a sectional-hyperbolic attracting set for a Hölder- $C^{1}$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^{1}$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.

Insensitivety to initial condition/prior in data assimilation for the case of the optimal filter and deterministic model

&lt;p&gt;Data assimilation is a term used to describe efforts to improve our knowledge&lt;br&gt;of a system by combining incomplete observations with imperfect models.&lt;br&gt;This is more generally known as filtering, which is &amp;#8217;optimal&amp;#8217; estimation of&lt;br&gt;the state of a system as it evolves over time, in the mean square sense. In&lt;br&gt;a Bayesian framework, the optimal filter is therefore naturally a sequence of&lt;br&gt;conditional probabilities of a signal given the observations and can be up-&lt;br&gt;dated recursively with new observations with Bayes&amp;#8217; formula. When, the&lt;br&gt;dynamics and observations errors are linear, this is equivalent to the Kalman&lt;br&gt;filter. In the nonlinear case, deriving an explicit form for the posterior dis-&lt;br&gt;tribution is in general not possible.&lt;br&gt;One of the important difficulties with applying the nonlinear filter in practice&lt;br&gt;is that the initial condition, the prior, is required to initialise the filtering.&lt;br&gt;However we are unlikely to know the correct initial distribution accurately&lt;br&gt;or at all. A filter is called stable if it is insensitive with respect to the&lt;br&gt;prior, that is, it converges to the same distribution, regardless of the initial&lt;br&gt;condition.&lt;br&gt;A body of work exists showing stability of the filter which rely on the stochas-&lt;br&gt;ticity of the underlying dynamics. In contrast, we show stability of the op-&lt;br&gt;timal filter for a class of nonlinear and deterministic dynamical systems and&lt;br&gt;our result relies on the intrinsic chaotic properties of the dynamics. We build&lt;br&gt;on the considerable knowledge that exists on the existence of SRB measures&lt;br&gt;in uniformly hyperbolic dynamical systems and we view the conditional prob-&lt;br&gt;abilities as SRB measures &amp;#8216;conditional on the observation&amp;#8217; which are shown&lt;br&gt;to be absolutely continuous along the unstable manifold. This is in line with&lt;br&gt;the result of Bouquet, Carrassi et al [1] regarding data assimilation in the&lt;br&gt;&amp;#8220;unstable subspace&amp;#8221;, where they show stability of the filter if the unstable&lt;br&gt;and neutral subspaces are uniformly observed.&lt;/p&gt;&lt;p&gt;[1] M. Bocquet et al. &amp;#8220;Degenerate Kalman Filter Error Covariances and&lt;br&gt;Their Convergence onto the Unstable Subspace&amp;#8221;. In: SIAM/ASA Jour-&lt;br&gt;nal on Uncertainty Quantification 5.1 (2017), pp. 304&amp;#8211;333. url: https:&lt;br&gt;//doi.org/10.1137/16M1068712.&lt;/p&gt;