arXiv subjects

Nikola P. Petrov

Publications and source records attributed to Nikola P. Petrov.

7 recordsLinked to original sources

arXiv2026

Convergence Rates for Semistochastic Processes

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

math.PR↗

arXiv2008

Boundaries of Siegel disks - numerical studies of their dynamics and regularity

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parameterization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parameterization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parameterization of the boundaries and the corresponding scaling exponents.

math.DS↗

arXiv2006

Regularity properties of critical invariant circles of twist maps, and their universality

We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier analysis method developed by R. de la Llave and one of the authors, we compute numerically the Holder regularities of these functions. Our computations show that -- withing their numerical accuracy -- these regularities are the same for the different maps studied. We discuss how our findings are related to some previous results: (a) to the constants giving the scaling behavior of the iterates on the critical invariant circle (discovered by Kadanoff and Shenker); (b) to some characteristics of the singular invariant measures connected with the distribution of iterates. Some of the functions studied have pointwise Holder regularity that is different at different points. Our results give a convincing numerical support to the fact that the points with different Holder exponents of these functions are interspersed in the same way for different maps, which is a strong indication that the underlying twist maps belong to the same universality class.

nlin.CD↗

arXiv2006

Energy Dissipation in Fractal-Forced Flow

The rate of energy dissipation in solutions of the body-forced 3-d incompressible Navier-Stokes equations is rigorously estimated with a focus on its dependence on the nature of the driving force. For square integrable body forces the high Reynolds number (low viscosity) upper bound on the dissipation is independent of the viscosity, consistent with the existence of a conventional turbulent energy cascade. On the other hand when the body force is not square integrable, i.e., when the Fourier spectrum of the force decays sufficiently slowly at high wavenumbers, there is significant direct driving at a broad range of spatial scales. Then the upper limit for the dissipation rate may diverge at high Reynolds numbers, consistent with recent experimental and computational studies of "fractal-forced'' turbulence.

physics.flu-dyn↗

arXiv2004

Variational bounds on the energy dissipation rate in body-forced shear flow

A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations.

physics.flu-dyn↗

arXiv2004

Dynamical Casimir effect in a periodically changing domain: A dynamical systems approach

We study the problem of the behavior of a quantum massless scalar field in the space between two parallel infinite perfectly conducting plates, one of them stationary, the other moving periodically. We reformulate the physical problem into a problem about the asymptotic behavior of the iterates of a map of the circle, and then apply results from theory of dynamical systems to study the properties of the map. Many of the general mathematical properties of maps of the circle translate into properties of the field in the cavity. For example, we give a complete classification of the possible resonances in the system, and show that small enough perturbations do not destroy the resonances. We use some mathematical identities to give transparent physical interpretation of the processes of creation and amplification of the quantum field due to the motion of the boundary and to elucidate the similarities and the differences between the classical and quantum fields in domains with moving boundaries.

quant-ph↗

arXiv2004

Low-wavenumber forcing and turbulent energy dissipation

A popular method of forcing the fluid in Direct Numerical Simulations of turbulence is to take the body force proportional to the projection of the velocity of the fluid onto its lowest Fourier modes, while keeping the injected external power constant. In this paper we perform a simple but rigorous analysis to establish bounds on the relationship between the energy dissipation rate and the resulting Reynolds number for this type of forcing. While this analysis cannot give detailed information of the energy spectrum, it does provide some indication of the balance of energy between the lower, directly forced, modes, and those excited by the cascade.

physics.flu-dyn↗