Kathleen Hoffman

NUMERICAL COMPUTATION OF CANARDS

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

The Forced van der Pol Equation II: Canards in the Reduced System ∗

This is the second in a series of papers about the dynamics of the forced van der Pol oscillator (J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2(2 003), pp. 1-35). The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in (J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2(2 003), pp. 1-35). We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system.

BIFURCATIONS OF RELAXATION OSCILLATIONS NEAR FOLDED SADDLES

Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow–fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one-parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, in which relaxation oscillations become homoclinic to a folded saddle.

On the derivation and tuning of phase oscillator models for lamprey central pattern generators

Using phase response curves and averaging theory, we derive phase oscillator models for the lamprey central pattern generator from two biophysically-based segmental models. The first one relies on network dynamics within a segment to produce the rhythm, while the second contains bursting cells. We study intersegmental coordination and show that the former class of models shows more robust behavior over the animal’s range of swimming frequencies. The network-based model can also easily produce approximately constant phase lags along the spinal cord, as observed experimentally. Precise control of phase lags in the network-based model is obtained by varying the relative strengths of its six different connection types with distance in a phase model with separate coupling functions for each connection type. The phase model also describes the effect of randomized connections, accurately predicting how quickly random network-based models approach the determinisitic model as the number of connections increases.

Methods for determining stability in continuum elastic-rod models of DNA

Elastic–rod models of DNA have offered an alternative method for studying the macroscopic properties of the molecule. An essential component of the modelling effort is to identify the biologically accessible, or stable, solutions. The underlying variational structure of the elastic–rod model can be exploited to derive methods that identify stable equilibrium configurations. We present two methods for determining the stability of the equilibria of elastic–rod models: the conjugate–point method and the distinguished–diagram method. Additionally, we apply these methods to two intrinsically curved DNA molecules: a DNA filament with an A–tract bend and a DNA minicircle with a catabolite gene activator protein binding site. The stable solutions of these models provide visual insight into the three–dimensional structure of the DNA molecules.

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values

Stability of n-covered Circles for Elastic Rods with Constant Planar Intrinsic Curvature

\sigma \le 1

An Extended Conjugate Point Theory with Application to the Stability of Planar Buckling of an Elastic Rod Subject to a Repulsive Self-Potential

at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and

Period Doubling Cascades in a Predator-Prey Model with a Scavenger

\sigma

The Bordered Operator and the Index, of a Constrained Critical Point

.Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter