Richard Haberman

Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems

Parameter-dependent equilibrium solutions are analyzed as the parameter slowly varies through critical values corresponding to a bifurcation or to a jump phenomena. At these critical times, interior nonlinear transition layers are necessary. Depending on the particular situation, local scaling analysis yields the first and a second Painleve transcendent among other generic equations. In specific cases the resulting boundary layer solutions either increase algebraically or explode (via a singularity). The algebraic growth corresponds to a smooth transition to a bifurcated equilibrium. When a jump phenomena is expected, an explosion can occur. In this case, the solution of first-order differential equations approaches the equilibrium, describing the slow evolution through such a jump. However, second-order differential equations have finite amplitude oscillations around the new equilibrium.

Resonantly coupled nonlinear evolution equations

A differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self‐modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse‐scattering problem.

The Modulated Phase Shift for Strongly Nonlinear, Slowly Varying, and Weakly Damped Oscillators

The phase shift and corresponding small frequency modulation for weakly dissipated nonlinear oscillators with slowly varying coefficients is calculated for the first time. This extends and corrects earlier work by Kuzmak, Luke, and Dobrokhotov and Maslov.

Separatrix crossing: time-invariant potentials with dissipation

Strongly nonlinear oscillations in a double-well potential that cross a separatrix due to dissipation are analyzed. Away from the separatrix the nearly periodic oscillations are found using standard asymptotic techniques that are known to fail near the separatrix. The solution near the separatrix is represented as a large sequence of perturbed solitary pulses. The asymptotic behavior of this sequence is matched (forward and backward in time) to the nearly periodic oscillations. In this manner, the oscillations in the post-crossing region are connected, for the first time, to initial conditions. In particular, we determine the amplitude and the phase of the nonlinear oscillator after crossing the separatrix and show them to be sensitively dependent on the initial conditions. The shift between the critical times associated with the slow variation theory before and after crossing the separatrix is derived.

Slow Passage through a Saddle-Center Bifurcation

Slowly varying conservative one-degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation. Away from unperturbed homoclinic orbits, strongly nonlinear oscillations are obtained using the method of averaging. A long sequence of nearly homoclinic orbits is matched to the strongly nonlinear oscillations before and after the slow passage. Usually solutions pass through the separatrix associated with the double homoclinic orbit before the saddle-center bifurcation occurs. For the case of slow passage through the special homoclinic orbit associated with a saddle-center bifurcation, solutions consist of a large sequence of nearly saddle-center homoclinic orbits connecting autonomous nonlinear saddle approaches, and the change in the action is computed. However, if the energy at one specific saddle approach is particularly small, then that one nonlinear saddle approach instead satisfies the nonautonomous first Painleve transcendent, in which case the solution usually either passes through the saddle-center homoclinic orbit in nearly the same way or makes a transition backwards in time to small oscillations around the stable center (though it is possible that the solution approaches the unstable saddle).

Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture

Strongly nonlinear oscillators under slowly varying perturbations (not necessarily Hamiltonian) are analyzed by putting the equations into the standard form for the method of averaging. By using the usual near-identity transformations, energy-angle (and equivalent action-angle) equations are derived using the properties of strongly nonlinear oscillators. By introducing a perturbation expansion, a differential equation for the phase shift is derived and shown to agree with earlier results obtained by Bourland and Haberman using the multiple scale perturbation method. The slowly varying phase shift is used (by necessity) to determine the boundary of the basin of attraction for competing stable equilibria, even though these averaged equations are known not to be valid near a separatrix (unperturbed homoclinic orbit).

Energy Bounds for the Slow Capture by a Center in Sustained Resonance

Critical trajectories for an autonomous nonlinear differential equation, perturbed by slowly varying coefficients, are analyzed in the case which Kevorkian has shown describes sustained roll resonance. The method of matched asymptotic expansions is used to calculate initial conditions for which a previously inaccessible center is captured. The results are significantly simplified using the energy dissipated along the critical trajectories.

Slow passage through homoclinic orbits for the unfolding of a saddle-center bifurcation and the change in the adiabatic invariant

Slowly varying, conservative, one degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation. At the bifurcation, a homoclinic orbit connects to a nonhyperbolic saddle point. Using averaging for strongly nonlinear oscillations, action is an adiabatic invariant before and after the slow passage of the homoclinic orbit. The homoclinic orbit is assumed to be crossed near to its creation in the saddle-center bifurcation, a dynamic unfolding. A large sequence of nearly homoclinic orbits with autonomous saddle approaches is matched to the strongly nonlinear oscillations valid before and after. Connection formulas are computed, determining the change in the action due to the slow passage through the unfolding of the saddle-center bifurcation. If the energy in one specific saddle region is particularly small, as occurs near the boundaries of the basin of attraction, then the solution in only that saddle region satisfies the nonautonomous Painleve I.

Wave-induced distortions of a slightly stratified shear flow: a nonlinear critical-layer effect

A slightly stratified shear flow is considered when the effects of nonlinearity, viscosity and thermal diffusivity are in balance in the critical layer. Finite amplitude essentially non-diffusive neutral waves exist only if the mean temperature, velocity and vorticity profiles are distorted such that small jumps in these quantities occur across the critical layer.

Slow Passage Through the Nonhyperbolic Homoclinic Orbit Associated with a Subcritical Pitchfork Bifurcation for Hamiltonian Systems and the Change in Action

Slowly varying conservative systems are analyzed in the case of a reverse subcritical pitchfork bifurcation in which two saddles and a center coalesce. Before the bifurcation there is a hyperbolic double-homoclinic orbit connecting a linear saddle point. At the bifurcation a double nonhyperbolic homoclinic orbit connects to a nonlinear saddle point. Strongly nonlinear oscillations obtained by the method of averaging are not valid near unperturbed homoclinic orbits. In the case in which the solution slowly passes through the nonhyperbolic homoclinic orbit associated with the subcritical pitchfork bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits connecting autonomous nonlinear saddle approaches. Solutions are captured into the left and right well. Phase jumps and the boundaries of the basins of attraction are computed. It is shown that the change in action in the slow passage through the nonhyperbolic homoclinic orbits is much larger than the known change in action f...