![]() Invariant-manifold theory also gives the existence of a centre-stable manifold W cs ( p ) and a centre-unstable manifold W cu ( p ) so that a centre manifold is obtained as the transverse intersection W c ( p ) = W cs ( p ) ⋔ W cu ( p ). Associated with the saddle-node equilibrium p at μ = 0 is its stable manifold W s ( p ), which consists of orbits that converge towards p exponentially in t as t → ∞, and its unstable manifold W u ( p ), which consists of orbits that converge towards p exponentially in t as t → − ∞. If this hypothesis is met, then the vector field on the one-dimensional centre manifold can be brought into normal form x ∙ c = b ( μ ) + a ( μ ) ( x c ) 2 + O ( | x c | 3 ), where a ( 0 ) ≠ 0 and b μ ( 0 ) ≠ 0. The unfolding is generic: 〈 w c, f μ ( p, 0 ) 〉 ≠ 0. The saddle-node equilibrium is not degenerate: 〈 w c, f u u ( p, 0 ) 〉 ≠ 0. ![]() The following conditions define a generic saddle-node bifurcation: (1) Hypothesis 2.5 Codimension-one Saddle-node Bifurcation Define v c and w c to be the right and left eigenvectors of the eigenvalue 0 of f u ( p, 0 ). In generic systems, homoclinic orbits to a saddle-node equilibrium occur as a codimension-one phenomenon. ![]() Centre manifolds and normal forms can then be used to study the local bifurcations near the equilibrium p, and we refer the reader to for their properties and various examples.įirst, consider the case where p is a saddle-node equilibrium: its linearization f u ( p, 0 ) has a simple real eigenvalue ν = 0 and no further eigenvalues on the imaginary axis. We now discuss various geometric notions that we shall use later when we review bifurcations of homoclinic orbits that converge to nonhyperbolic equilibria.įrom now on, let p be a nonhyperbolic equilibrium of u ∙ = f ( u, 0 ), so that f u ( p, 0 ) has at least one eigenvalue on the imaginary axis. In systems with additional structure, such as reversibility or a Hamiltonian structure, this may be typical or at least of low codimension. Like hyperbolic equilibria, nonhyperbolic equilibria may admit homoclinic solutions. Ale Jan Homburg, Björn Sandstede, in Handbook of Dynamical Systems, 2010 2.2 Homoclinic orbits to nonhyperbolic equilibria
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