On Sinaĭ Billiards on Flat Surfaces with Horns

We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

Rogue waves in the KdV-type models

<p>In this study, we investigate the rogue-wave-type phenomena in the physical systems described by the Korteweg-de Vries (KdV)-like equation in the form $ u_t + [u^m \sgn{u}]_x + u_{xxx} = 0 $ with the arbitrary real coefficient $m>0$. The periodic waves (sinusoidal or cnoidal) described by this equation have been shown to suffer from the modulational instability if $m \ge 3$; the modulational growth results in the formation of rogue waves similar to the Peregrine, Kuznetsov-Ma or Akhmediev breathers known for the nonlinear Schrodinger equation. In this work we focus on the rogue wave occurrence in ensembles of soliton-type waves. First of all, the characteristics of the solitary waves are investigated depending on the power $m$. The existence of solitary waves with exponential tails, as well as algebraic solitons and compactons has been shown for different ranges of the parameter $m$ values. Their energetic stability is discussed. Two solitary wave/breathers interactions are studied as elementary acts of the soliton/breather turbulence. It is demonstrated that the property of attracting solitons/breathers is a necessity condition for the formation of rogue waves. Rigorous results are obtained for the integrable versions of the KdV-type equations. Series of numerical simulations of the rogue wave generation has been conducted for different values of $m$. The obtained results are applied to the problems of surface and internal waves in the ocean, and to elastic waves in the solid medium.</p><p>The research is supported by the RNF grant 19-12-00253.</p>

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

Consider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

Forecast for the second Covid-19 wave based on the improved SIR model with a constant ratio of recovery to infection rate

We start out by deriving simple analytic expressions for all measurable amounts of cases and fatalities during a pandemic evolution exhibiting multiple waves, described by the semi-time SIR model. The approximant shares all relevant features with the exact solution, including time and position of the peak of daily new infections, as well as the asymptotic behaviors at small and large times. We derive exact analytic expressions for the early doubling time, late half decay time, and a half-early peak law, characterizing the dynamical evolution. We show, in particular, how the asymmetry of the first epidemic wave and its exponential tails are affected by the initial conditions; a feature that has no analogue in the all-time SIR model. We apply the approach to available data from different continents. Our analysis reveals that the immunity is very strongly increasing during the 2nd wave, while it was still at a very moderate level of a few percent in several countries at the end of the first wave. The wave-specific SIR parameters describing the infection and recovery rates we find to behave in a similar fashion, while their ratio k was decreasing only by a about 5% for most countries. Still, an apparently moderate change of k can have significant consequences for the relevant numbers like the final amount of infected or deceased population. As we show, the probability for an additional wave is however low in several countries due to the fraction of immune inhabitants at the end of the 2nd wave, irrespective the currently ongoing vaccination efforts. We compare with alternate approaches.

Inducing schemes for multi-dimensional piecewise expanding maps

<p style='text-indent:20px;'>We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.</p>

Continuously heated granular gas of elongated particles

Some years ago, Harth et al. experimentally explored the steady state dynamics of a heated granular gas of rod-like particles in microgravity [K. Harth et al. Phys. Rev. Lett. 110, 144102 (2013)]. Here, we report numerical results that quantitatively reproduce their experimental findings and provide additional insight into the process. A system of sphero-cylinders is heated by the vibration of three flat side walls, resulting in one symmetrically heated direction, one non-symmetrically heated direction, and one non-heated direction. In the non-heated direction, the speed distribution follows a stretched exponential distribution $$p(\upsilon )\, \propto \,{\rm{exp}}\left( { - {{\left( {{{\left| \upsilon \right|} \mathord{\left/ {\vphantom {{\left| \upsilon \right|} C}} \right. \kern-\nulldelimiterspace} C}} \right)}^{1.5}}} \right)$$. In the symmetrically heated direction, the velocity statistics at low speeds is similar but it develops pronounced exponential tails at high speeds. In the non-symmetrically heated direction (not accessed experimentally), the distribution also follows $$p(\upsilon )\, \propto \,{\rm{exp}}\left( { - {{\left( {{{\left| \upsilon \right|} \mathord{\left/ {\vphantom {{\left| \upsilon \right|} C}} \right. \kern-\nulldelimiterspace} C}} \right)}^{1.5}}} \right)$$ , but the velocity statistics of rods moving toward the vibrating wall resembles the indirectly excited direction, whereas the velocity statistics of those moving away from the wall resembles the direct excited direction.