Localized structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of length scales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behavior can be obtained using a variational method, for two systems. I...

Opto-mechanical chaotic oscillation of an on-chip resonator is excited by the radiation-pressure nonlinearity. Continuous optical input, with no external feedback or modulation, excites chaotic vibrations in very different geometries of the cavity (both tori and spheres) and shows that opto-mechanical chaotic oscillations are an intrinsic property of optical microcavities. Measured phenomena in...

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic α, prescription (α=0, 1/2, and 1, respectively, correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state p(st)(x) for a class of constitutive relations between drift and diffusion and show that it has a q-exponential form, p(st)(x)=N(q)[1-(1-q)βV...

We present results of physical experiments where we measure the autocorrelation function (ACF) and the spectral linewidth of the basic frequency of a spiral chaotic attractor in a generator with inertial nonlinearity both without and in the presence of external noise. It is shown that the ACF of spiral attractors decays according to an exponential law with a decrement which is defined by the ph...

We develop an exponentially convergent distributed algorithm to minimize a sum of nonsmooth cost functions with set constraint. The constraint generally leads the nonlinearity in algorithms, and results difficulties derive exponential rate. In this article, we remove consensus constraints by exact penalty method, then propose projected subgradient virtue differential inclusion differentiated pr...

Explanation of exponential growth in self-reproduction is an important step toward elucidation of the origins of life because optimization of the growth potential across rounds of selection is necessary for Darwinian evolution. To produce another copy with approximately the same composition, the exponential growth rates for all components have to be equal. How such balanced growth is achieved, ...

Exponential family statistical distributions, including the well-known Normal, Binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distribut...

We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy l...

We prove a conjecture by De Giorgi, which states that global weak solutions of nonlinear wave equations such as w + |w|p−2w = 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations. These functionals, which are integrals in space-time of a convex Lagrangian, contain an exponential weight with a parameter ε, and the initial data of the wave equa...

We investigate entire radial solutions of the semilinear biharmonic equation ∆u = λ exp(u) in Rn, n ≥ 5, λ > 0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s = log |x|. Next, we pro...