A generalization of Vinograd's theorem for dynamical systems

The theory of dynamical systems has been expanded by the introduction of local dynamical systems [10, 4, 9] and local semidynamical systems [1]. Using integral curves of autonomous ordinary differential equations to illustrate these generalizations, we find that, roughly, the integral curves form a local dynamical system if solutions exist and are unique without requiring existence for all time, and the integral curves form a local semidynamical system if solutions exist and are unique in the positive sense but need not exist for all positive time. In addition to autonomous ordinary differential equations, the enlarged theory of dynamical systems has applications to nonautonomous ordinary differential equations, certain partial differential equations, functional differential equations, and Volterra Integral equations [9, 1, 2, 8], respectively. All of these have metric phase spaces. Since many dynamic considerations are invariant to reparameterizations, it is of interest to known when a local dynamical (or semidynamical) system can be reparameterized to yield a “global” dynamical (or semidynamical) system. For autonomous ordinary differential equations, Vinograd [7] has shown that the local dynamical system on an open subset ofRⁿ formed by integral curves is isomorphic (in the sense of Nemytskii and Stepanov) to a global dynamical system. In an extensive study of isomorphisms, Ura [12] has expanded the Gottschalk-Hedlund notion of an isomorphism and restated Vinograd's result in terms of a reparameterization. In this paper we study the problem of finding a global dynamical (or semidynamical) system which is isomorphic to a given local system. A necessary and sufficient condition is found which is then used to show that the Vinograd result holds on metric spaces.     

Taylor series method for dynamical systems with control: Convergence and error estimates

To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize. We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision, while the aggregative Taylor series method achieves the best computational time. The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of ODEs describing the optimal flight of a spacecraft from the Earth to the Moon. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

An elementary approach to the analysis of exact solutions for the Navier-Stokes stagnation flows with slips

We deal with the exact solutions of the Navier-Stokes equations for stagnation flows with slips. The problem becomes the solvability of certain third-order ordinary differential equations (ODEs). Reducing the order of ODEs, we exhibit another elementary proof of the existence and asymptotic behavior of solutions. Numerical investigations are also provided. Received: 14 August 2003     

A Theory of Complexity for Continuous Time Systems

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

Generalized notions of symmetry of ODEs and reduction procedures

This paper describes the notion of σ‐symmetry, which extends the one of λ‐symmetry, and its application to reduction procedures of systems of ordinary differential equations (ODEs) and of dynamical systems (DS) as well. We also consider orbital symmetries, which give rise to a different form of reduction of DS. Finally, we discuss how DS can be transformed into higher order ODEs, and how these symmetry properties of the DS can be transferred into reduction properties of the corresponding ODEs. Many examples illustrate the various situations. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

求解微分方程初值问题的一种弧长法

对于连续介质力学问题中导出的微分方程初值问题,常常具有解奇异性,如不连续、Ｓｔｉｆ性质或激波间断·本文通过在相应空间,引入一个或数个弧长参数变量,克服解的奇异性·对于常微分方程组引入弧长参数变量后,奇异性得以消除和削弱,应用一般的解常微分方程组的方法（如Ｒｕｎｇｅ＿Ｋｕｔａ法）求解·对于偏微分方程引入弧长参数变量后,在相应的空间离散成常微分方程组,用解奇异性常微分方程组相同的方法即可求解·本文给出了两个算例     

Hayman’s classical conjecture on some nonlinear second-order algebraic ODEs

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second-order nonlinear algebraic ordinary differential equations (ODEs). We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the classical conjecture proposed by Hayman in 1996.     

Lattice-valued fuzzy Turing machines: Computing power, universality and efficiency

In this paper we study fuzzy Turing machines with membership degrees in distributive lattices, which we called them lattice-valued fuzzy Turing machines. First we give several formulations of lattice-valued fuzzy Turing machines, including in particular deterministic and non-deterministic lattice-valued fuzzy Turing machines (l-DTMcs and l-NTMs). We then show that l-DTMcs and l-NTMs are not equivalent as the acceptors of fuzzy languages. This contrasts sharply with classical Turing machines. Second, we show that lattice-valued fuzzy Turing machines can recognize n-r.e. sets in the sense of Bedregal and Figueira, the super-computing power of fuzzy Turing machines is established in the lattice-setting. Third, we show that the truth-valued lattice being finite is a necessary and sufficient condition for the existence of a universal lattice-valued fuzzy Turing machine. For an infinite distributive lattice with a compact metric, we also show that a universal fuzzy Turing machine exists in an approximate sense. This means, for any prescribed accuracy, there is a universal machine that can simulate any lattice-valued fuzzy Turing machine on it with the given accuracy. Finally, we introduce the notions of lattice-valued fuzzy polynomial time-bounded computation (lP) and lattice-valued non-deterministic fuzzy polynomial time-bounded computation (lNP), and investigate their connections with P and NP. We claim that lattice-valued fuzzy Turing machines are more efficient than classical Turing machines.     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.     

What is a universal computing machine?

A computer is classically formalised as a universal Turing machine or a similar device. However over the years a lot of research has focused on the computational properties of dynamical systems other than Turing machines, such cellular automata, artificial neural networks, mirrors systems, etc.In this paper we propose a unifying formalism derived from a generalisation of Turing’s arguments. Then we review some of universal systems proposed in the literature and show that are particular case of this formalism. Finally, we review some of the attempts to understand the relation between dynamical and computational properties of a system.     

Remarks on Third-Order ODEs Relevant to the Kuramoto-Sivashinsky Equation

We are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.     

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

The nonexistence of spurious solutions to discrete,two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which are independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented.     

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.