Quadratic Dynamical Systems and Algebras

Abstract

Quadratic dynamical systems come from differential or discrete systems of the form Ẋ = Q(X) or X(k+1)=Q(X(k)), where Q:Rn→Rn is homogeneous of degree 2; i.e., Q(αX) = α2Q(X) for all α∈R, X∈Rn. Defining a bilinear mapping β:Rn × Rn→Rn by β(X, Y) ≔ 12[Q(X+Y)−Q(X)−Q(Y)], we view XY≡β(X, Y) as a multiplication, and thus consider A=(Rn, β) to be a commutative, nonassociative algebra. The quadratic systems are then studied with the general theme that the structure of the algebras helps determine the behavior of the solutions. For example, semisimple algebras give a decoupling of the original system into systems occurring in simple algebras, and solvable algebras give solutions to differential systems via linear differential equations; the general three-dimensional example of the latter phenomenon is described. There are many classical examples and the scope of quadratic systems is large; every polynomial system can be embedded into a higher dimensional quadratic system such that solutions of the original system are obtained from the quadratic system. For differential systems, nilpotents of index 2 (N2=0) are equilibria and idempotents (E2=E) give ray solutions. The origin is never asymptotically stable, and the existence of nonzero idempotents implies that the origin is actually unstable. Nonzero equilibria are not hyperbolic, but can be studied by standard algebra techniques using nondegenerate bilinear forms as Lyapunov functions. Periodic orbits lie on "cones." They cannot occur in dimension 2 or in power-associative algebras. No periodic orbit can be an attractor but "limit cycles" (invariant cones) can exist. Automorphisms of the algebra A leave equilibria, periodic orbits, and domains of attraction invariant. Also, explicit solutions can be given by the action of automorphisms on an initial point; the general three-dimensional example of this is described. Thus if there are sufficient automorphisms, Hilbert′s sixteenth problem in R3 has the following answer: if the periodic orbits of fixed period are isolated, then there is only one cone of periodic solutions; this cone is almost an attractor. For discrete systems there are many similarities to the differential systems. For example, orbits can be given by automorphisms, and again, the general three-dimensional example of this is described. However, distinctions become more obvious using algebras; for example, if the algebra A is nilpotent, then for the differential system, the solutions are unbounded, but for the discrete system, the trajectories iterate to zero in A; also idempotents E2=E are the fixed points for the discrete system, and E is unstable if there exist suitable nilpotents N2=0. The interplay between algebras and dynamical systems can solve old problems, but more importantly, it can create new opportunities in both areas.