Chaos, Solitons & Fractals Vol. 4, No. 11, pp. 1969-I984, 1.994 Elsevier Science Ltd Pergamon Printed in Great Britain. 0960-0779(94)00160-X Integrability and Attractors JOSEPH L. McCAULEY Physics Department, University of Houston, Houston, TX 77204-5506, USA (Received 16 June 1994) Abstract--We discuss global and local integrability and also attractors from a physicist's standpoint. A local Lie method is used to form an analytic and geometric picture of integrability via conservation laws. The starting point is Lie's proof that two-dimensional flows, even dissipative ones, have a conservation law and so are integrable. Through conservation laws, integrable higher-dimensional flows reduce to the two-dimensional case. We consider a class of models defined by the Euler- Lagrange equations for a set of nonintegrable velocities (nonholonomic coordinates). The destruction of the global conservation laws of this conservative system by the inclusion of linear damping and constant external driving leads to self-confinement and attractors in phase space. In particular, the Lorenz model belongs to this class and can be seen as a damped, driven symmetric top. 1. INTEGRABLE FLOWS IN PHASE SPACE We study integrability by following the consequences of an elementary theorem by Sophus Lie. According to Bell [1], Lie showed that every one-parameter infinitesimal transforma- tion acting on two variables defines a one-parameter group, and that any such group has an invariant. This means that two-dimensional flows in phase space, whether conservative or dissipative, have a conservation law. This fact is not pointed out in standard texts on differential equations or mechanics. To understand Lie's theorem, it is only necessary to establish the language of phase flows along with a dictionary to translate back and forth between the language of groups [2] and the language of differential equations [3]. The discussion that follows is heuristic in the spirit of Lie [4] and Klein [5] (and in the spirit of physics) rather than rigorous in the spirit of modern abstract mathematics [6]. By a flow, we mean an autonomous system of n (generally coupled and nonlinear) differential equations dxi -- V/(x1 ..... Xn) (1) dt where the solutions are finite at all finite times (any and all singularities are confined to the imaginary time axis [3]). Phase space is the n-dimensional space indicated by the local coordinates (xl .... , x,). By a conservative flow we mean one that preserves the volume element dff2 = dxa • •. dx, in phase space, so that n ° V" V = z Oxi = O. (2) i=10xi 1970 J.L. McCAuLEV In an arbitrary conservative system, the volume element is the only quantity that is left invariant by the flow. That is, in a completely arbitrary conservative flow there are generally no conservation laws that are point functions. Conservation laws are functions of the phase space variables and the time, Gg(x 1..... x n, t) = constant, that satisfy d Gi 3 G i -- V" VG i -b -- = O. (3) dt ~t By a nonconservative or 'dissipative' system we mean any system where V.V fails to vanish identically over all of phase space. A typical dissipative system has no invariant at all, neither the phase volume element nor a solution of the partial differential equation (3). The integration of the flow (1) studied as a collection of infinitesimal transformations (~Xi ~- V/(X1 ..... Xn)f~t (4) defines a one parameter continuous group of transformations Xi(t ) = lPi(Xl(tO) ..... Xn(to) , t -- to) (5) from coordinates (xl(to) ..... xn(to)) to (xl(t) ..... x~(t)) as the group parameter is varied from to to t. In the language of differential equations, (5) is the solution at a later time t and (x1(to) ..... xn(to)) is the initial condition at to. The time t is a canonical group parameter, which means that it obeys an additive parameter-combination law. Noncanon- ical group parameters obey nonlinear parameter-combination laws. For a one parameter transformation on two variables, a canonical parameter can always be found by integration [7]. In terms of the canonical parameter, the infinitesimal transformations of the group are defined by the differential equations dxl - Vl(xl, x2) dt dx2 - g2(xl, x2) (6) dt which also describe a flow in the (x~, x2) phase space. Lie's theorem is now easy to demonstrate heuristically. By rewriting the flow equations in the form dt = dxl/V 1 = dx2/V2 (7) we can obtain the time-independent differential form V2(x 1, x2)dxl -- VI(X 1, x2)dx2 = 0. (8) Every differential form in only two variables is either integrable or else has an integrating factor M(Xl, x2) that makes it integrable, so that we can integrate the differential form M(V2 dXl - VI dx2) = dG (8b) to obtain G(Xa, x2)= C, which is a conservation law. For example, this means that even two-dimensional Newtonian flows where the mechanical energy E is not constant have a conservation law. Before continuing, we pause to state the condition under which the conservation law for a dissipative system exists, which is simply the condition that an integrating factor M exists. We can rewrite the differential form 1/2 dXl - V1 dx2 = 0 as the one-dimensional equation dx2 V2(Xl, x2) - - F(xl, x2) (9) dXl Vl(xl, x2) Integrability and attractors 1971 whose solution satisfies the integral equation x2 = X2o + f(s, x2(s))ds. (9b) Xlo According to Picard [8], this equation can be solved by the method of successive approximations so long as F satisfies a Lipshitz condition. A sufficient condition is that that ratio of the components of the velocity in phase V2/V1 must be at least once continuously differentiable with respect to the variable x~. This requires that we avoid integrating over regions that contain equilibria or other zeros of Vs. The resulting solution of the integral equation yields x2 as a function of xl. A qualitatively similar result follows from solving the conservation law G(xl, x2) = C to find x2 = f(x~, C), which is just the statement that the conservation law determines x2 as a function of Xl and C. In particular, differentiating the conservation law once yields dG = 3G dx~ + 3G dx2 = 0, (10) 3xl 9x2 and comparison with the differential form V2 dxl - V1 dx2 = 0 then shows that / - 3G/3---G-G = V2/V1. (lla) 3Xl/3X2 This last equation is satisfied only if V 1 : -M 3G and 172 = M olG, (llb) 3x2 ~Xl which guarantees the existence of the integrating factor M. In other words, the existence of the conservation law follows from satisfying Picard's conditions for a unique solution of the differential equation (9). This guarantees, by differentiation, that the integrating factor M exists. Except in a few special cases, there is no systematic way to construct the integrating factor M(x~, x2) even though it is guaranteed formally to exist. Notice that the condition that the flow is phase volume preserving, ~V 2 3V 1 - --, (llc) 9x2 3Xl is just the condition that the differential form V2(xl, x2)dxl- Vl(Xl, x2)dx2 = 0 is exact without any need for an integrating factor. This means that the velocity field of a conservative two-dimensional flow can always be obtained from a stream function G: V = (V1, V2) : ( ~x29G'9~1. (12) By inspection, the conservative system is Hamiltonian with Hamiltonian G and canonically conjugate variables (q, p)= (x2, &), but the collapse of an integrable conservative flow into canonical form is peculiar to two dimensions and generally is not true for integrable conservative flows in higher dimensions: in particular, every three-dimensional integrable flow is noncanonical although it may still be Hamiltonian on a Poisson manifold that is not symplectic [10]. As an example of a conservation law for a dissipative system, consider the damped simple harmonic oscillator de' + /3:~ + wax = 0. (13) 1972 J.L. McCAULEY With x I = X and x 2 -~- dx/dt the corresponding differential form (fix2 + a~zxl)dxa + x2 dx2 = 0 (14) is not exact but is guaranteed to have an integrating factor M that yields a time-independ- ent conservation law G(x, dx/dt) = constant via the condition dG - M((13x2 + O~Xl)~l + x2~2) = 0. (15) dt The conservation law for this specific case was originally discovered and constructed by Palmore et al. [9]. In all cases, the conservation law G(x, dx/dt)= C is simply the equation that describes that family of streamlines: different constants C correspond to different streamlines within the same family. For the damped oscillator, the streamlines are a family of spirals. Lie's result explains qualitatively why two-dimensional dissipative as well as conservative flows have conservation laws: the condition C --- G(Xlo , x2o ) fixes the constant C in terms of the two initial conditions Xio at time t = to and is simply the equation of the family of streamlines of a flow, G(Xl, x2) = C, in the phase plane. Each streamline corresponds to a different value of the constant C. For a dissipative system, the conservation law is singular at a source or sink and this nonanalyticity gives rise to nontrivially different constants C for different streamlines [10]. If we can solve G(Xl, x2) = C to find the streamline equations in the form xz = f2(Xl, C) and/or xl = fl(x2, C), which requires that the implicit function theorem [11] is satisfied, then we can rewrite at least one of the differential equations in the form dG =0 dt = dxi/vi(xi, C). (16) The problem is therefore 'integrable', meaning that the complete solution can be reduced to two independent integrations.