By Alain Goriely
This necessary booklet examines qualitative and quantitative tools for nonlinear differential equations, in addition to integrability and nonintegrability conception. ranging from the assumption of a continuing of movement for easy structures of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical results. Integrability thought is approached from diversified views, first when it comes to differential algebra, then by way of complicated time singularities and at last from the point of view of part geometry (for either Hamiltonian and non-Hamiltonian systems). As common structures of differential equations can't be precisely solved, the booklet stories the several notions of nonintegrability and indicates the best way to turn out the nonexistence of actual strategies and/or a continuing of movement. ultimately, nonintegrability idea is associated with dynamical platforms conception by way of displaying how the valuables of whole integrability, partial integrability or nonintegrability should be concerning common and abnormal dynamics in part house.
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Extra info for Integrability and nonintegrability of dynamical systems
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.