x) =- .* -co,(x,c) = 0, (v = 1, . . ., n) (1)be a system of n autonomous differential equations of first order . The co, are assumed tobe polynomials in x. In addition they may contain one or more parameters Ck which aredenoted collectively as c = (cl , c2. . . . ). One of the most prominent features governing thesolution set of the autonomous system (1) is the chaotic behaviour which may occur insome regions of the parameter space as opposed to a highly regular behaviour which maybe characteristic for some other region. It is one of the basic problems connected withsystems of this kind to characterise these regions (Lichtenberg & Lieberman, 1983). Thereis no general answer to this question. The situation is completely different, however, iffirst integrals can be found. By definition a function f is a first integral for (1) if its totalderivative with respect to time vanishes, i.e. if it satisfiesdf =0 (2)dtunder the constraints (1). If fdoes not depend on t explicitly this condition may bewritten ascov8f(x)= 0 (3)
[1]
Marek Kus,et al.
Integrals of motion for the Lorenz system
,
1983
.
[2]
F. Schwarz,et al.
Symmetries and first integrals for dissipative systems
,
1984
.
[3]
W B Steeb,et al.
Continuous symmetries of the Lorenz model and the Rikitake two-disc dynamo system
,
1982
.
[4]
Joel Moses,et al.
Solutions of systems of polynomial equations by elimination
,
1966,
CACM.
[5]
David Y. Y. Yun,et al.
On algorithms for solving systems of polynomial equations
,
1973,
SIGS.
[6]
David Y. Y. Yun,et al.
On solving systems of algebraic equations via ideal bases and elimination theory
,
1981,
SYMSAC '81.