Douglas S. Shafer

The 1: −q resonant center problem for certain cubic Lotka–Volterra systems

Abstract Necessary conditions and distinct sufficient conditions are derived for the system x ˙ = x ( 1 - a 20 x 2 - a 11 xy - a 02 y 2 ) , y ˙ = y ( - q + b 20 x 2 + b 11 xy + b 02 y 2 ) to admit a first integral of the form Φ ( x , y ) = x q y + ⋯ in a neighborhood of the origin, in which case the origin is termed a 1 : - q resonant center. Necessary and sufficient conditions are obtained for odd q , q ⩽ 9 ; necessary conditions, most of which are also sufficient, are obtained for even q , q ⩽ 8 . Key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions, and the theory of Darboux first integrals to show sufficiency.

Multiple bifurcation in a predator–prey system with nonmonotonic predator response

A model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.

Tutorial: Symbolic computation and the cyclicity problem for singularities

We show how methods of computational commutative algebra are employed to investigate the local 16th Hilbert Problem, which is to find an upper bound on the number of limit cycles that can bifurcate from singularities in families of polynomial systems of differential equations on R^2, and is one step in a program for solving the full 16th Hilbert Problem. We discuss an extension of a well-known theorem, and illustrate the concepts and methods with concrete examples.

Time-Reversibility in Two-Dimensional Polynomial Systems

We characterize the set of all time-reversible systems within a particular family of complex polynomial differential equations in two complex dimensions. These results are a generalization to the complex case of theorems of Sibirsky for real systems. We also give an efficient computational algorithm for finding this set. An interconnection of time-reversibility and the center problem is discussed as well.

Singularities of gradient vectorfields in R3

Let N be a neighborhood of (0, 0,O) in 9 3 and V = U + W a C’ function, I > k + 3, from N to 9, where U is a homogeneous polynomial of degree k + 1, and W vanishes together with all its derivatives through order k + 1 at 0. Let g be a smooth Riemannian metric on .R3 and let X = grad, V. If k = 1, then the critical point of V at 0 is non-degenerate if and only if U,, U,,, and U, vanish together only at (0, 0, 0), in which case the singularity of X at (0, 0,O) is hyperbolic. Then the Hartman-Grobman theorem implies that X is topologically equivalent near (0, 0,O) to its linear part; i.e., the topological equivalence class of X is determined by its first non-zc:o jet. We denote this X Xi. For k > 1, let X, denote the homogeneous polynomial vectorfield of degree k with the same k-jet at (0, 0,O) as X (Xk is the “homogeneous part” of X). Let x denote the vectorfield on a neighborhood of S2 x (0) in S* X .5? obtaining from X by polar blowing up. The main result is:

Topological conjugacy of singularities of vector fields

ROUGHLY speaking, a smooth vector field is finitely determined for topological conjugacy at a singularity if its conjugacy class contains some finite Taylor approximation. Finite determinacy is one criterion for the existence of conjugacies between smooth planar vector fields. This criterion can fail at a singularity only if the vector field is either P-flat or not sufficiently smooth. In both situations we construct specific examples of singularities. not rotation points, that are topologically equivalent but not conjugate. In addition, we determine a useful necessary and sufficient condition for the conjugacy of a large class of singularities. This allows for a simple characterization of the resulting conjugacy classes.

Any two quadratic cycles can be connected with a smooth arc

Abstract It is known that every limit cycle of a quadratic system can be connected to a Hopf bifurcation. We show that every pair of cycles (i.e., periodic orbits) in quadratic systems (having the same orientation) can be connected to one another by an arc of cycles, no intermediate one of which lies in the period annulus of a center, or within an algebraic curve of degree less than four.

Complete integrability and time-reversibility of some 3-dim systems

Abstract In this work we study complete integrability and time-reversibility of three dimensional polynomial systems with a singularity at the origin at which the linear part has a zero eigenvalue and a pair of eigenvalues that are the negatives of one another. An approach to finding systems within a given parametric family admitting two independent analytic local first integrals in a neighborhood of the origin is proposed and applied to a family of cubic systems.

Geometry of Cycles In Quadratic Systems

This paper is a study of the affine and euclidean differential geometry of cycles of quadratic systems. While the euclidean curvature must always be strictly positive, the affine curvature can take either sign, although with certain restrictions. Every quadratic cycle has exactly six affine vertices, but the number of euclidean vertices can vary, not just from cycle to cycle, but for the same cycle under a linear coordinate transformation. We prove that an upper bound on the number of euclidean vertices over all non-circular quadratic cycles is twelve, and provide evidence that a sharp upper bound is six.

Topological hyperbolicity and the Coleman Conjecture for diffeomorphisms

Abstract A definition of topological hyperbolicity is presented which applies to fixed points of diffeomorphisms as well as to singularities of vector fields. Its strength relative to other definitions of topological hyperbolicity is explored. In particular, a theorem of F. W. Wilson validating the Coleman Conjecture concerning topological equivalence of flows is strengthened, and its corresponding statement for topological conjugacy shown false. An analogue of the Coleman Conjecture for diffeomorphisms is shown to be false in every case.