Continuum Hamiltonian Hopf Bifurcation I

Abstract

Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid models is considered. These models have linear dynamics with discrete spectra, when linearized about homogeneous equilibria, and these spectra have counterparts to the steady state and Hamiltonian Hopf bifurcations when equilibrium parameters are varied. Examples of fluid sound waves and plasma and gravitational streaming are treated in detail. Next, using these 1 + 1 examples as a guide, a large class of 2 + 1 Hamiltonian systems is introduced, and Hamiltonian bifurcations with continuous spectra are examined. It is shown how to attach a signature to such continuous spectra, which facilitates the description of the continuous Hamiltonian Hopf bifurcation. This chapter lays the groundwork for Kreȋn-like theorems associated with the CHH bifurcation that are more rigorously discussed in our companion chapter [1].

Variational necessary and sufficient stability conditions for inviscid shear flow

“…Due to the property W (Ω) = W (Ω), the eigenvalues always exist as pairs of growing (ω j ) and damping (ω j ) ones when Im ω j > 0, a spectral property of Hamiltonian systems (cf. [18][19][20]). For the purpose of showing the existence or nonexistence of such eigenvalues, we will frequently use the following Lemma.…”

“…(i) A consequence of E(ω)Φ(x, ω) = 0 is the following identity (see the Appendix of [43]): Unfortunately, this argument is not always true especially for nonmonotonic profiles of U (x) (see Ref. [40,41,43] for mathematical justification in the shear flow context and [18,19] for a discussion of k = 0 bifurcations in the Vlasov context). In this work, we simply assume the monotonicity (A2) and verify Tollmien's argument as follows.…”

“…Due to the property W (Ω) = W (Ω), the eigenvalues always exist as pairs of growing (ω j ) and damping (ω j ) ones when Im ω j > 0, a spectral property of Hamiltonian systems (cf. [18][19][20]).…”

“…C. Signature of Q Now, let us substitute the expression (19) into the quadratic form Q = ξ, Hξ , which is a constant of motion for the equation (7). By using the property of the resolvent (Ω − kL * ) −1 (see Ref.…”

“…A version of the quadratic form Q was previously given in [11], but it was not used to obtain sufficient conditions for instability. We note in passing that the discovery of signature for the continuous spectrum has also led to rigorous Kre ȋn-like theorems [18][19][20], where discrete eigenvalues emerge from the continuous spectrum, termed Continuum Hamiltonian Hopf bifurcations. (See also [21,22] on infinite-dimensional Hamiltonian systems and many other contributions in the recent book [23].…”