Two parallel, relativistic electron streams with different velocities support unstable, exponentially growing space-charge waves, and efficient electron bunching may be achieved in the submillimeter wavelength range. Injection of the prebunched streams into a static, periodic (wiggler) magnetic field enhances the intensity of the parametrically excited, back-scattered electromagnetic wave. Calc...

The response of a coupled array of nonlinear oscillators to parametric excitation is calculated in the weak nonlinear limit using secular perturbation theory. Exact results for small arrays of oscillators are used to guide the analysis of the numerical integration of the model equations of motion for large arrays. The results provide a qualitative explanation for a recent experiment @Buks and R...

We report experiments and model simulations of patterns of parametrically excited capillary ripples in a large aspect-ratio cell with thin horizontal layer of viscous fluid subjected to sinusoidal vertical oscillations. We found stable rotating spirals with different topological charges in the parameter range where steady straight rolls were observed previously. Formation of multiarmed spirals ...

An experimental study of surface waves parametrically excited by vertical vibrations is presented. The shape of the eigenmodes in a closed vessel, and the importance of the free-surface boundary conditions, are discussed. Stability boundaries, wave amplitude, and perturbation characteristic time of decay are measured and found to be in agreement with an amplitude equation derived by symmetry. T...

Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics e...

We investigate the dynamics of a delayed nonlinear Mathieu equation: ẍ + (δ + εα cos t)x + εγ x = εβx(t − T ) in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of averaging (valid for small ε) is used to obtain a slow flow that is analyzed for stability and bifurcations. We show that the 2:1...