A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media

This paper considers a class of space fractional partial differential equations (FPDEs) that describe gas pressures in fractured media. First, the well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the considered FPDEs are investigated. Then, the reference tracking problem is studied to track the pressure gradient at a downstream location of a channel. This requires manipulation of gas pressure at the downstream location and the use of pressure measurements at an upstream location. To achiever this, the backstepping approach is adapted to the space FPDEs. The key challenge in this adaptation is the non-applicability of the Lyapunov theory which is typically used to prove the stability of the target system as, the obtained target system is fractional in space. In addition, a backstepping adaptive observer is designed to jointly estimate both the system's state and the disturbance. The stability of the closed loop (reference tracking controller/observer) is also investigated. Finally, numerical simulations are given to evaluate the efficiency of the proposed method.Comment: 37 pages, 9 figure

Pattern selection models: From normal to anomalous diffusion

“Pattern formation and selection is an important topic in many physical, chemical, and biological fields. In 1952, Alan Turing showed that a system of chemical substances could produce spatially stable patterns by the interplay of diffusion and reactions. Since then, pattern formations have been widely studied via the reaction-diffusion models. So far, patterns in the single-component system with normal diffusion have been well understood. Motivated by the experimental observations, more recent attention has been focused on the reaction-diffusion systems with anomalous diffusion as well as coupled multi-component systems. The objectives of this dissertation are to study the effects of superdiffusion on pattern formations and to compare them with the effects of normal diffusion in one-, and multi-component reaction-diffusion systems. Our studies show that the model parameters, including diffusion coefficients, ratio of diffusion powers, and coupling strength between components play an important role on the pattern formation. Both theoretical analysis and numerical simulations are carried out to understand the pattern formation in different parameter regimes. Starting with the linear stability analysis, the theoretical studies predict the space of Turing instability. To further study pattern selection in this space, weakly nonlinear analysis is carried out to obtain the regimes for different patterns. On the other hand, numerical simulations are carried out to fully investigate the interplay of diffusion and nonlinear reactions on pattern formations. To this end, the reaction-diffusion systems are solved by the Fourier pseudo-spectral method. Numerical results show that superdiffusion may substantially change the patterns in a reaction-diffusion system. Different superdiffusive exponents of the activator and inhibitor could cause both qualitative and quantitative changes in emergent spatial patterns. Comparing to single-component systems, the patterns observed in multi-component systems are more complex”--Abstract, page iv

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction-diffusion systems

International audienceIn this paper, the problem of fixed-time stabilization of constant-parameter reaction-diffusion partial differential equations by means of continuous boundary time-varying feedbacks is considered. Moreover, the time of convergence can be prescribed in the design. The design of time-varying feedbacks is carried out based on the backstepping approach. Using a suitable target system with a time varying-coefficient, one can state that the resulting kernel of the backstepping transformation is time-varying and rendering the control feedback to be time-varying as well. Explicit representations of the kernel solution in terms of generalized Laguerre polynomials and modified Bessel functions are derived. The fixed-time stability property is then proved. A simulation example is presented to illustrate the main results