Experimental investigation of unsteady separation in the rotor-oscillator flow

Visualisations of various types of flow separation are presented in an experimental set-up that translates a rotating cylinder parallel to a wall. Particle image velocimetry is used to measure the two velocity components in a plane perpendicular to the cylinder where the flow is two-dimensional. To spatially resolve the flow close to the wall, a high-viscosity fluid is used. For a periodic translation, the fixed separation is compared to the theory of Haller (J. Fluid Mech., vol. 512, 2004, pp. 257–311), while for non-periodic translations, a method is proposed to extract the moving separation point captured by a Lagrangian saddle point, and its finite-time unstable direction (separation profiles). Intermediate cases are also presented where both types of separation, fixed and moving, are either present simultaneously or appear successively. Some results issued from numerical simulations of an impinging jet show that all the cases observed in the rotor-oscillator flow are not restricted to high-viscosity fluid motions but may also occur within any vortical flow.

Thermodynamics of the Katok map

We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2)110(3) 1979, 529–547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$, $t\neq 1$, where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$.

Lyapunov theorems for exponential dichotomies in Hilbert spaces

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

Maximizing the Statistical Diversity of an Ensemble of Bred Vectors by Using the Geometric Norm

It is shown that the choice of the norm has a great impact on the construction of ensembles of bred vectors. The geometric norm maximizes (in comparison with other norms such as the Euclidean one) the statistical diversity of the ensemble while at the same time it enhances the growth rate of the bred vector and its projection on the linearly most unstable direction (i.e., the Lyapunov vector). The geometric norm is also optimal in providing the least fluctuating ensemble dimension among all the spectrum of norms studied. The results are exemplified with numerical integrations of a toy model of the atmosphere (the Lorenz-96 model), but these findings are expected to be generic for spatially extended chaotic systems.

STOCHASTIC STABILITY OF NON-UNIFORMLY HYPERBOLIC DIFFEOMORPHISMS

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

The dynamic system method and the traps

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

The dynamic system method and the traps

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.