Toward Solution of Matrix Equation X=Af(X)B+C

This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation $X=Af(X) B+C$ with $f(X) =X^{\mathrm{T}},$ $f(X) =\bar{X}$ and $f(X) =X^{\mathrm{H}},$ where $X$ is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of $W=\mathcal{A}W\mathcal{B}+\mathcal{C}$ where the dimensions of the coefficient matrices $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are the same as those of the original equation. Closed-form solutions of equation $X=Af(X) B+C$ can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation $X=Af(X) B+C$. Necessary and sufficient conditions are established to guarantee the convergence of the iterations

The Effects of a Rapidly-Fluctuating Random Environment on Systems of Interacting Species

Some models of interacting species in a random environment are analyzed. Approximate solutions of the stochastic differential or delay-differential equations describing the systems are obtained, on the assumption that the random environment is fluctuating rapidly

Six-vertex model and non-linear differential equations I. Spectral problem

In this work we relate the spectral problem of the toroidal six-vertex model's transfer matrix with the theory of integrable non-linear differential equations. More precisely, we establish an analogy between the Classical Inverse Scattering Method and previously proposed functional equations originating from the Yang-Baxter algebra. The latter equations are then regarded as an Auxiliary Linear Problem allowing us to show that the six-vertex model's spectrum solves Riccati-type non-linear differential equations. Generating functions of conserved quantities are expressed in terms of determinants and we also discuss a relation between our Riccati equations and a stationary Schr\"odinger equation.Comment: 42 pages, 3 figure

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in t-direction. Hence, a numerical method would have to use infinitely many points.\ud \ud To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using Radial Basis Functions