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Geometric phases, reduction and LiePoisson structure for the resonant threewave interaction
 Physica D
, 1998
"... Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These resul ..."
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Cited by 16 (5 self)
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Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the threewave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.
1981] A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced
 Horseshoes in Perturbations on Hamiltonian Systems With Two Degrees of Freedom,” Commun
, 1981
"... This paper delineates a class of timeperiodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and ..."
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Cited by 11 (3 self)
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This paper delineates a class of timeperiodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form ˙x = f0(x) + εf1(x, t), where ˙x = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam. 1
Parabolic resonances in 3 degree of freedom nearintegrable Hamiltonian systems
 Physica D
, 2002
"... Perturbing an integrable 3 degree of freedom (d.o.f.) Hamiltonian system containing a normally parabolic 2torus which is mresonant (m = 1 or 2) creates a parabolic mresonance (mPR). PRs of different types are either persistent or of low codimension, hence they appear robustly in many applicatio ..."
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Cited by 10 (7 self)
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Perturbing an integrable 3 degree of freedom (d.o.f.) Hamiltonian system containing a normally parabolic 2torus which is mresonant (m = 1 or 2) creates a parabolic mresonance (mPR). PRs of different types are either persistent or of low codimension, hence they appear robustly in many applications. Energy–momenta bifurcation diagram is constructed as a tool for studying the global structure of 3 d.o.f. nearintegrable systems. A link between the diagram shape, PR and the resonance structure is found. The differences between the dynamics appearing in 2 and 3 d.o.f. systems exhibiting PRs are studied analytically and numerically. The numerical study demonstrates that PRs are an unavoidable source of large and fast
Branches of Stable ThreeTori Using Hamiltonian Methods in Hopf Bifurcation on a Rhombic Lattice
 of Systems
, 1996
"... This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4 . The normal form has a symmetry group given by the semidirect product of D2 with T 2 \Theta S 1 . The Hamiltonian part of the normal form is comple ..."
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Cited by 10 (10 self)
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This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4 . The normal form has a symmetry group given by the semidirect product of D2 with T 2 \Theta S 1 . The Hamiltonian part of the normal form is completely integrable and may be analyzed using a system of invariants. The idea of the paper is to perturb relative equilibria in this singular Hamiltonian limit to obtain new three frequency solutions to the full normal form for parameter values near the Hamiltonian limit. The solutions obtained have fully broken symmetry, that is, they do not lie in fixed point subspaces. The methods developed in this paper allow one to determine the stability of this new branch of solutions. An example shows that the branch of threetori can be stable. 1 Introduction A standard approach in the bifurcation analysis of spatiallyextended systems (such as RayleighBenard convection in an infinite plane) is to res...
Normal forms for threedimensional parametric instabilities
 Physica D
, 1994
"... We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ɛ, λ), where ɛ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the result ..."
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Cited by 9 (8 self)
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We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ɛ, λ), where ɛ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by threedimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or offcenter displacement. 1
Geometry and control of threewave interactions
 in The Arnoldfest
, 1997
"... The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to ..."
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Cited by 4 (1 self)
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The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigidbody or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasiphasematching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.
On Perturbed Oscillators in 111 Resonance: The Case of Axially Symmetric Cubic Potentials
, 2001
"... Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a onedegreeoffreedom system. We use a special set of actionangle variabl ..."
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Cited by 3 (0 self)
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Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a onedegreeoffreedom system. We use a special set of actionangle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We illustrate the results by analysing the family of (arbitrary) axially symmetric cubic potentials. 1 Introduction One of the few methods that are available to study Hamiltonian systems is to find an integrable system that is close to it and to consider the former as a perturbation of the latter. In case the integrable system is nondegenerate, the flow of this system makes the phase space a ramified torus bundle. For instance, in three degrees of ...
The 1:±2 Resonance
, 2007
"... On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In p ..."
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Cited by 1 (1 self)
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On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In particular we show that for the indefinite case (1:−2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is nondegenerate (i.e. the Kolmogorov nondegeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a by product of our analysis of the fequency map we obtain another proof of fractional monodromy in the 1:−2 resonance. 1
AN INVARIANT MANIFOLD THEORY FOR ODES AND ITS APPLICATIONS
, 909
"... Abstract. For a system of ODEs defined on an open, convex domain U containing a positively invariant set Γ, we prove that under appropriate hypotheses, Γ is the graph of a Cr function and thus a Cr manifold. Because the hypotheses can be easily verified by inspecting the vector field of the system, ..."
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Abstract. For a system of ODEs defined on an open, convex domain U containing a positively invariant set Γ, we prove that under appropriate hypotheses, Γ is the graph of a Cr function and thus a Cr manifold. Because the hypotheses can be easily verified by inspecting the vector field of the system, this invariant manifold theory can be used to study the existence of invariant manifolds in systems involving a wide range of parameters and the persistence of invariant manifolds whose normal hyperbolicity vanishes when a small parameter goes to zero. We apply this invariant manifold theory to study three examples and in each case obtain results that are not attainable by classical normally hyperbolic invariant manifold theory.
Mathematical Sciences HP Laboratories Bristol
, 1998
"... threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered and shown to be compatible. POIsson reduction is performed using the method of invariants and geom ..."
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threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered and shown to be compatible. POIsson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for Instance, nonlinear optics. Some of the general structures presented in the latter part of ~IS paper are implicit in the litet:atur~; our purpose IS to put the threewave Interaction In the modem setting of geometric mechanics and to explore some new things, such as integrability, in thIS context.