Backlund transformations for a class of systems of differential equations

Backlund Transformations and Superposition formulae are given for a class of systems of nonlinear differential equations for matrix valued functions ofn independent variables. Whenn=2 the equations reduce to the wave equation, the sine-Gordon equation, the Laplace equation and the elliptic sinh-Gordon equation.The first author is partially supported by CNPq.The second author is partially supported by CAPES and CNPq.     

On the transformations of the dynamical equations

In this issue we bring to the reader’s attention a translation of Levi-Civita’s work “Sulle trasformazioni delle equazioni dinamiche”. This paper, written by Levi-Civita at the onset of his career, is remarkable in many respects. Both the main result and the method developed in the paper brought the author in line with the greatest mathematicians of his day and seriously influenced the further progress of geometry and the theory of integrable systems. Speaking modern language the main result of his paper is the deduction of the general geodesic equivalence equation in invariant form and local classification of geodesically equivalent Riemannian metrics in the case of arbitrary dimension, i.e., metrics having the same geodesics considered as unparameterized curves (this classification problem was formulated by Beltrami in 1865). Levi-Civita’s work produced a great impact on further development of the theory of geodesically equivalent metrics and geodesic mappings, and still remains one of the most important tools in this area of differential geometry. In this paper the author uses a new method based on the concept of Riemannian connection, which later has been also referred to as the Levi-Civita connection. This paper is truly a pioneering work in the sense that the real power of covariant differentiation techniques in solving a concrete and highly nontrivial problem from the theory of dynamical systems was demonstrated. The author skillfully operates and weaves together many of the most advanced (for that times) algebraic, geometric and analytic methods. Moreover, an attentive reader can also notice several forerunning ideas of the method of moving frames, which was developed a few decades later by E. Cartan. We hope that the reader will appreciate the style of exposition as well. This work, focused on the essence of the problem and free of manipulation with abstract mathematical terms, is a good example of a classical text of the late 19th century. Owing to this, the paper is easy to read and understand in spite of some different notation and terminology. The Editorial Board is very grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments (see marginal notes at the end of the text, p. 612).     

On a certain class of dynamical systems on the torus with rational rotation number

For system (1), the authors solve the problem of the existence of trajectories given by the linear equation y ≡ μx + y ₀ (mod 2π), μ ≥ 0, y ₀ ∈ ℝ, and in the case where such trajectories exist, they perform a complete qualitative study of the system on the two-dimensional torus under certain conditions. Translated From Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics And Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Moderate deviations for randomly perturbed dynamical systems

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form X_n=f(X_n−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of the process staying in a small tube around the deterministic system. Using these, explicit formulae for the asymptotics of exit times are obtained. Results are specified for the case when the dynamical system is periodic, and imply stability of such systems. Finally, results are applied to the model of density-dependent branching processes.     

The estimate of the time of motion of certain dynamical systems

An estimate of the time necessary for the phase points of a dynamical system to reach a specified finite domain, containing an asymptotically stable solution, from any initial position belonging to the specified domain is obtained with the sole assumption that the derivative of the Lyapunov function for autonomous second-order systems and for certain higher-order systems has a negative sign.     

On certain alternating boundary-layer solutions to singularly perturbed parabolic equations

The solution to a special singularly perturbed parabolic problem with a right-hand independent of the spatial variable is studied numerically and analytically.     

Global stabilization of a certain class of nonlinear dynamical systems using state detection

This note is concerned with the stabilization of control systems using an estimated state feedback. The global stabilization problem for a relatively broad class of nonlinear plants is discussed. Moreover, using the “input to state stability” property introduced by Sontag [1–4] and detectability condition, we show that the system can be globally asymptotically stable using a state detection.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.