John David Crawford

Amplitude expansions for instabilities in populations of globally-coupled oscillators

We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes.

Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings

Abstract The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling; the effect of a stochastic temporal variation in the frequencies is also included. The Fokker-Planck equation for the coupled Langevin system is reduced to a kinetic equation for the oscillator distribution function. Instabilities of the phase-incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes. Depending on the coupling, the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected. A detailed analysis of these singularities to all orders in the normal form expansion is presented. Physically, the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization. These predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions.

Application of the method of spectral deformation to the Vlasov-Poisson system

Abstract The Vlasov equation is studied using the method of spectral deformation, a technique developed in the theory of Schrodinger operators. The method associates the linearized Vlasov operator L with a family of operators L (θ) which depend analytically on the complex parameter θ. The analysis of L (θ) leads to a novel solution to the linear initial value problem, a new eigenfunction expansion which can be applied to nonlinear problems, and a deeper understanding of the relation between the spectrum of L and Landau damping. For example, it follows from the theory of L (θ) that the embedded eigenvalues of L , which characteristically occur at the threshold of a linear instability, are generically simple. Extending the deformation construction to include the nonlinear terms leads to a family of evolution equations depending on the parameter θ. This family is shown to be time reversal invariant in a suitably generalized sense. The analyticity properties required of Vlasov solutions if they are to correspond to solutions of the new equations for complex θ are described. The eigenfunction expansion for L (θ) is applied to derive equations for the nonlinear evolution of electrostatic waves. This derivation and the resulting amplitude equations are unusual in that the standard assumptions of weak nonlinearity and separated time scales are not used. When these assumptions are made, the familiar form of the amplitude equations is recovered.

Hidden symmetries of parametrically forced waves

The dynamics of parametrically excited surface waves in square containers reveal the effects of symmetry at several levels. In addition to the expected square symmetry D4 admitted by the fluid equations and the boundary conditions, there are hidden translational and rotational symmetries that further constrain the linear and nonlinear behaviour of the fluid. As a result one finds unexpected degeneracies among the linear wave frequencies and unexpected branches of nonlinear solutions in the bifurcation equations for the surface waves. These additional symmetries are not obvious since they are not symmetries of the square container and consequently do not preserve the boundary conditions of the problem. The author can include them in a theoretical analysis by extending the fluid equations of the original problem to larger domains with greater symmetry; in this enlarged problem the previously hidden symmetries now enter in the usual way. Among other prerequisites, this extension depends on the square container having straight sidewalls.

Normal forms for driven surface waves: boundary conditions, symmetry, and genericity

Abstract The parametric excitation of standing waves in a vertically oscillated fluid can be studied by analyzing the period-doubling bifurcation in the associated stroboscopic map. When the fluid container has symmetry Γ the stroboscopic map will exhibit this symmetry and this in turn significantly affects the structure of the bifurcation problem. For both square and circular containers the normal forms appropriate to the reduced maps for the critical mode amplitudes are discussed. The experiments of Simonelli and Gollub with square containers strongly suggest that the effective symmetry of the problem is larger than the obvious geometric symmetry of the boundaries. For the ideal fluid model of Benjamin and Ursell with Neumann boundary conditions (NBC) one can show that the problem does indeed have a larger symmetry because it embeds in a model on a larger domain with periodic boundary conditions (PBC); the original NBC are then realized as a symmetry constraint on the solutions of this extended model. The additional symmetries introduced in this way lead to a large number of effects that would otherwise be non-generic including critical eigenspaces carrying reducible representations of Γ and reduced maps on these eigenspaces that have symmetries larger than Γ. These extra symmetries relate in-phase and out-of-phase mixed modes and lead to new branches to pure modes. The case of circular geometry is also discussed as it does not allow this extension to PBC and thus provides an instructive contrast with the subtleties found for square geometry.

Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: no distinguished parameter

Abstract The Hopf bifurcation in the presence of O(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented. No distinguished parameter is assumed. The universal unfoldings reveal the existence of both 2-tori and 3-tori; corresponding to quasiperiodic waves with two and three independent frequencies, respectively.

Phosphaturic Effect of Cortisone in Normal and Parathyroidectomized Rats.

Summary 1) The effect of cortisone on serum inorganic phosphorus concentration and urinary phosphorus excretion has been investigated in normal and parathyroidectomized rats. 2) It has been found that cortisone exerts a phosphaturic action which is not dependent upon the presence of the parathyroid glands nor accompanied by elevation of serum phosphorus concentration.

Amplitude equations for electrostatic waves: Universal singular behavior in the limit of weak instability

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the unstable manifold of the equilibrium F0(v). The mode eigenvalue arises from a simple zero of the dielectric ek(z); as the linear growth rate γ vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude ρ(t) is studied using an expansion in ρ. As γ→0+, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: ρ(t)≡γ2r(γt). This renders the theory finite and also indicates that the electric field exhibits trapping scaling E∼γ2. These singularities and scalings are independent of the specific F0(v) considered. The asymptotic dynamics of r(τ) can depend on F0 only through exp iξ where dek/dz=‖ek’‖exp−iξ/2. Similar results also hold for the electric field and distribution function.An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the unstable manifold of the equilibrium F0(v). The mode eigenvalue arises from a simple zero of the dielectric ek(z); as the linear growth rate γ vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude ρ(t) is studied using an expansion in ρ. As γ→0+, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: ρ(t)≡γ2r(γt). This renders the theory finite and also indicates that the electric field exhibits trapping scaling E∼γ2. These singularities and scalings are independent of the specific F0(v) considered. The asymptotic dynamics of r(τ) can depend on F0 only through exp iξ where dek/dz=‖ek’‖exp−iξ/2. Similar results also hold for the electric field and distribution function.

On degenerate Hopf bifurcation with broken O(2) symmetry

A symmetry-breaking Hopf bifurcation in an O(2)-equivariant system generally produces a branch of standing waves and two branches of oppositely propagating travelling waves. This generic bifurcation assumes three non-degeneracy conditions on the cubic terms of the Poincare-Birkhoff normal form. When these conditions fail more complicated behaviour accompanies the bifurcation; in particular one finds secondary bifurcations of quasiperiodic waves. For these degenerate bifurcations, the effects of perturbations which break the reflection symmetry are considered. The perturbed system retains a residual SO(2) symmetry. Qualitatively these perturbations have three effects: (1) they split the double multiplicity eigenvalues to that the travelling waves bifurcate separately, (2) they perturb the primary standing wave branches to secondary branches of modulated waves and (3) they produce new steady-state bifurcations along the modulated wave branches.

Application of spectral deformation to the Vlasov-Poisson system. II. Mathematical results

This paper presents a mathematical description of the linearized Vlasov–Poisson operator Lk acting on a family of Banach spaces Xp, related to L p (R), and the application of the method of spectral deformation to this model. It is shown that a type‐A analytic family of operators Lk(θ), θ∈C, Lk(0) =Lk can be associated with Lk. By means of this family, the Landau damped modes of the plasma are identified as the spectral resonances of Lk. Existence and uniqueness of solutions to the initial‐value problem for the evolution equation ∂ν g =Lk(θ)g is proven. An expansion of any solution to the initial‐value problem (with sufficiently smooth initial data) is obtained in terms of the eigenfunctions of Lk(θ) and a spectral integral over the essential spectrum. This is applied to derive an expansion for solutions to the Vlasov equation in which the Landau damped portions of the distribution function are manifestly exhibited. A self‐contained discussion of the spectral deformation method and an extension of it to ce...