No Arabic abstract
Some scaling properties for classical light ray dynamics inside a periodically corrugated waveguide are studied by use of a simplified two-dimensional nonlinear area-preserving map. It is shown that the phase space is mixed. The chaotic sea is characterized using scaling arguments revealing critical exponents connected by an analytic relationship. The formalism is widely applicable to systems with mixed phase space, and especially in studies of the transition from integrability to non-integrability, including that in classical billiard problems.
The chaotic low energy region of the Fermi-Ulam simplified accelerator model is characterised by use of scaling analysis. It is shown that the average velocity and the roughness (variance of the average velocity) obey scaling functions with the same characteristic exponents. The formalism is widely applicable, including to billiards and to other chaotic systems.
In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.
A problem of diffraction of a symmetrical transverse magnetic mode $ text{TM}_{0l} $ by an open-ended cylindrical waveguide corrugated inside is considered. A depth and a period of corrugations are supposed to be much less than the wavelength and the waveguide radius. Therefore a corrugated waveguide wall can be described in terms of equivalent boundary conditions, i.e. a corresponding impedance boundary condition can be applied. Both vacuum case and the case of uniform dielectric filling of the waveguide is considered. The diffraction problem is solved using the modified tayloring technique in Jones formulation. Solution of the Wiener-Hopf-Fock equation of the problem is used to obtain an infinite linear system for reflection coefficients, the latter can be solved numerically using the reduction technique.
The Nikolaevskiy model for pattern formation with continuous symmetry exhibits spatiotemporal chaos with strong scale separation. Extensive numerical investigations of the chaotic attractor reveal unexpected scaling behavior of the long-wave modes. Surprisingly, the computed amplitude and correlation time scalings are found to differ from the values obtained by asymptotically consistent multiple-scale analysis. However, when higher-order corrections are added to the leading-order theory of Matthews and Cox, the anomalous scaling is recovered.
The scaling behavior of the maximal Lyapunov exponent in chaotic systems with time-delayed feedback is investigated. For large delay times it has been shown that the delay-dependence of the exponent allows a distinction between strong and weak chaos, which are the analogy to strong and weak instability of periodic orbits in a delay system. We find significant differences between scaling of exponents in periodic or chaotic systems. We show that chaotic scaling is related to fluctuations in the linearized equations of motion. A linear delay system including multiplicative noise shows the same properties as the deterministic chaotic systems.