Search results for " DYNAMICAL SYSTEM"
showing 10 items of 188 documents
Genericity of dimension drop on self-affine sets
2017
We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
On a rough perturbation of the Navier-Stokes system and its vorticity formulation
2019
We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous rando…
Fair immunization and network topology of complex financial ecosystems
2023
The aftermath of the recent financial crisis has shown how expensive and unfair the stabilization of financial ecosystems can be. The main cause is the level of complexity of financial interactions that poses a problem for regulators. We provide an analytical framework that decomposes complex ecosystems in both their overall level of instability and the contribution of institutions to instability. These ingredients are then used to study the pathways of the ecosystems towards stability by means of immunization schemes. The latter can be designed to penalize institutions proportionally to their contribution to instability, and therefore enhance fairness. We show that fair immunization scheme…
Quantitative ergodicity for some switched dynamical systems
2012
International audience; We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd × E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.
Hybrid Modelling and Control of a Class of Power Converters With Triangular-Carrier PWM Inputs
2021
In this paper, a new control design procedure for a class of power converters based on hybrid dynamical systems theory is presented. The continuous-time dynamics, as voltage and current signals, and discretetime dynamics, as the on-off state of the switches, are captured with a hybrid model. This model avoids the use of averaged and approximated models and includes the PWM as well as the sample-and-hold mechanism, commonly used in the industry. Then, another simplified hybrid system, whose trajectories match with the original one, is selected to design the controller and to analyse stability properties. Finally, an estimation of the chattering in steady state of the voltage and current sign…
Random attractors for stochastic lattice systems with non-Lipschitz non-linearity
2012
In this paper we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor.
Structural Stability
2020
The notion of structural stability was first introduced by the Russian math- ematicians Alexandr Andronov and Lev Pontryagin (cf. Andronov and Potryangin 1937). However, there are traces of such a concept in the work of the French math- ematician Henry Poincaré (cf. Poincaré 1880). In more recent years, interesting developments about structural stability included writings of important math- ematicians like Mauricìo Peixoto (cf. at least Peixoto 1960), Stephen Smale (cf. at least Smale 1971) and René Thom (1972, 1980) (see structural morpho- dynamics). From an intuitive point of view, structural stability refers to a particular systemic property known as robustness. Put in general terms, a s…
Seifert manifolds admitting partially hyperbolic diffeomorphisms
2017
We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.
Surface homeomorphisms with zero dimensional singular set
1998
We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set is totally disconnected, then f is topologically conjugate to a conformal transformation.
Dirac physical measures for generic diffeomorphisms
2016
We prove that, for a $C^1$ generic diffeomorphism, the only Dirac physical measures with dense statistical basin are those supported on sinks.