Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation

AR Champneys; PD Woods

Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation

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Abstract

This paper considers an unfolding of a degenerate reversible $1-1$ resonance (or Hamiltonian Hopf) bifurcation for four-dimensional systems of time-reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become two imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q_2=0) that determines whether the bifurcation is super- or sub-critical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur.J.Mech.B-Fluids 15, (1996) 367-393), given a sign of the coefficient of a higher-order term (q_40 and -1

Original language	English
Publication status	Published - 1998

Bibliographical note

Additional information: Preprint of a paper later published by Elsevier Science (1999), Physica D - Nonlinear Phenomena, 129 (3-4), pp.147-170, ISSN 0167-2789

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http://hdl.handle.net/1983/441

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title = "Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation",

abstract = "This paper considers an unfolding of a degenerate reversible $1-1$ resonance (or Hamiltonian Hopf) bifurcation for four-dimensional systems of time-reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become two imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q_2=0) that determines whether the bifurcation is super- or sub-critical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur.J.Mech.B-Fluids 15, (1996) 367-393), given a sign of the coefficient of a higher-order term (q_40 and -1 ",

author = "AR Champneys and PD Woods",

note = "Additional information: Preprint of a paper later published by Elsevier Science (1999), Physica D - Nonlinear Phenomena, 129 (3-4), pp.147-170, ISSN 0167-2789 ",

year = "1998",

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T1 - Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation

AU - Champneys, AR

AU - Woods, PD

N1 - Additional information: Preprint of a paper later published by Elsevier Science (1999), Physica D - Nonlinear Phenomena, 129 (3-4), pp.147-170, ISSN 0167-2789

PY - 1998

Y1 - 1998

N2 - This paper considers an unfolding of a degenerate reversible $1-1$ resonance (or Hamiltonian Hopf) bifurcation for four-dimensional systems of time-reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become two imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q_2=0) that determines whether the bifurcation is super- or sub-critical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur.J.Mech.B-Fluids 15, (1996) 367-393), given a sign of the coefficient of a higher-order term (q_40 and -1

AB - This paper considers an unfolding of a degenerate reversible $1-1$ resonance (or Hamiltonian Hopf) bifurcation for four-dimensional systems of time-reversible ordinary differential equations (ODEs). This bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become two imaginary pairs. The degeneracy occurs via the vanishing of a normal form coefficient (q_2=0) that determines whether the bifurcation is super- or sub-critical. Of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits. A partial unfolding of such solutions already occurs in the work of Dias and Iooss (Eur.J.Mech.B-Fluids 15, (1996) 367-393), given a sign of the coefficient of a higher-order term (q_40 and -1

M3 - Working paper and Preprints

BT - Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation

ER -