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This tutorial survey presents a method for computing the Lyapunov quantities for Liénard systems of differential equations using symbolic manipulation packages. The theory is given in detail and simple working MATLAB and Maple programs are listed in this chapter. In recent years, the author has been contacted by many researchers requiring more detail on the algorithmic method used to compute focal...
We consider a simple computational approach to estimating the cyclicity of centers in various classes of planar polynomial systems. Among the results we establish are confirmation of Żoł¸dek’s result that at least 11 limit cycles can bifurcate from a cubic center, a quartic system with 17 limit cycles bifurcating from a non-degenerate center, and another quartic system with at least 22 limit cycles...
In this article, the definition of isochronous center at infinity is given and the center conditions and the isochronous center conditions at infinity for a class of differential systems are investigated. By a transformation, infinity is taken to the origin and therefore properties at infinity can be studied with the methods developed for finite critical points. Using the computations of singular...
We consider the class of polynomial differential equations ẋ = Pn(x, y)+Pn+m(x, y)+Pn+2m(x, y)+Pn+3m(x, y), ẏ = Qn(x, y)+Qn+m(x, y)+Qn+2m(x,y)+Qn+3m(x, y), for n,m ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i. Inside this class we identify new subclasses of Darboux integrable systems...
We characterize the set of all time-reversible systems within a particular family of complex polynomial differential equations in two complex dimensions. These results are a generalization to the complex case of theorems of Sibirsky for real systems. We also give an efficient computational algorithm for finding this set. An interconnection of time-reversibility and the center problem is discussed...
In the present paper, based on our earlier work, we propose a systematic method for symbolically computing the Lyapunov characteristic exponents, briefly LCE, of n-dimensional dynamical systems. First, we analyze in mathematics the LCE of n-dimensional dynamical systems. In particular, as an example, we discuss the LCE of the Lorenz systems. Then, to do the above task, a framework on representation...
Many practical dynamic models contain complicated nonlinearities that make it difficult to investigate the distribution and qualitative properties of equilibria, which are actually the basic information for further discussion on bifurcations. In this paper effective methods of symbolic computation are introduced for two nonlinear systems.
This paper aims to investigate attractive regions of operating points for power systems by applying singular perturbation analysis. A time-scale decomposition is performed to illustrate how the critical model can be identified with reduced-order systems and how bifurcation phenomena can be explained with such low order systems. The slow dynamics and fast dynamics including bifurcation conditions and...
In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the system at the singular points. A method for computing the algebraic multiplicity using Newton polygon is also presented.
There is a reasoning strategy, which is an incremental computation, used in symbolic and algebraic approach to differential equations. The center-focus problem can be solved by using this reasoning strategy. In algebraic approach to automated reasoning, the construction of polynomial ideals is at the heart. For polynomials with a known fixed number of variables, the problem of constructing polynomial...
We describe usage of the normal form method and corresponding computer algebra packages for building an approximation of local periodic solutions of nonlinear autonomous ordinary differential equations (ODEs). For illustration a number of systems are treated.
First order algebraic differential equations are considered. A necessary condition for a first order algebraic differential equation to have a rational general solution is given: the algebraic genus of the equation should be zero. Combining with Fuchs’ conditions for algebraic differential equations without movable critical point, an algorithm is given for the computation of rational general solutions...
An algorithm for factoring differential systems in characteristic p has been given by Cluzeau in [7]. It is based on both the reduction of a matrix called p-curvature and eigenring techniques. In this paper, we generalize this algorithm to factor partial differential systems in characteristic p. We show that this factorization problem reduces effectively to the problem of simultaneous reduction of...
Differential modules are modules over rings of linear (partial) differential operators which are finite-dimensional vector spaces. We present a generalization of the Beke-Schlesinger algorithm that factors differential modules. The method requires solving only one set of associated equations for each degree d of a potential factor.
We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws of nonlinear systems of partial differential...
A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (system) if that is possible. Infinite parameter conservation laws can be computed, for example, with the computer...
An algorithm to explicitly compute polynomial conservation laws for nonlinear evolution equations (either uniform in rank or not) is introduced and a software package CONSLAW written in Maple to automate the computation is developed. CONSLAW can construct the polynomial conservation laws for polynomial partial differential equations automatically. Furthermore, some new integrable systems can be filtered...
Generalized differential resultants of algebraic ODEs are introduced and relations between generalized differential resultant systems and differential elimination are shown.
The characteristic set method of polynomial equations-solving is naturally extended to the differential case, which gives rise to an algorithmic method for solving arbitrary systems of algebrico-differential equations. The existence of “good bases” of the associated algebrico-differential ideals is also studied in this way. As an illustration of the method, the Devil problem of Pommaret is studied...
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