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By Napalkov V.V.

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192). It allows us to find conditions on the parameters of a vector field for the existence of timeindependent first integrals and provides an explicit method for their computation. The calculations 58 CHAPTER 2. , 1990a; Moulin-Ollagnier, 1990). ∂x in R3 together with their corresponding systems of ODEs x˙ = f (x), and x˙ = g(x). ] between two such vector fields is defined by the vector field whose components are 3 [f , g]i = (δf g − δg f )i = ( j=1 ∂gi ∂fi fj − gj ). ∂x I = 0. Then, the three vector fields f , g and [f , g] are linearly dependent for all points x ∈ R3 .

X v) 4. Find a first integral I = I(u, v) of the first order equation du dv = G(u, v). In order to illustrate the techniques and difficulties associated with this method we apply it on a simple example. 25 The Lotka-Volterra ABC system. c) where x = (x1 , x2 , x3 ) and A, B, C are parameters. 119) where L is a 3 × 3 matrix with constant coefficients.

Wi 42 CHAPTER 2. 13 Non-existence of scale invariant solutions. Not all scale-invariant systems have scale-invariant solutions. b) x˙ 2 = + x22 )2 , is clearly homogeneous of degree d = 5 and hence scale-invariant with w = (−1/4, −1/4). However, 1 there is no scale-invariant solution since the ansatz x = αt− 4 implies α1 = α2 = 0. 2 Homogeneous and weight-homogeneous decompositions Most vector fields are neither homogeneous nor weight-homogeneous. For instance if the diagonal terms of the linear part of a nonlinear vector field do not vanish identically, then the system is not weight-homogeneous.

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