Slow Passage through a Saddle-Center Bifurcation

Slowly varying conservative one-degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation. Away from unperturbed homoclinic orbits, strongly nonlinear oscillations are obtained using the method of averaging. A long sequence of nearly homoclinic orbits is matched to the strongly nonlinear oscillations before and after the slow passage. Usually solutions pass through the separatrix associated with the double homoclinic orbit before the saddle-center bifurcation occurs. For the case of slow passage through the special homoclinic orbit associated with a saddle-center bifurcation, solutions consist of a large sequence of nearly saddle-center homoclinic orbits connecting autonomous nonlinear saddle approaches, and the change in the action is computed. However, if the energy at one specific saddle approach is particularly small, then that one nonlinear saddle approach instead satisfies the nonautonomous first Painleve transcendent, in which case the solution usually either passes through the saddle-center homoclinic orbit in nearly the same way or makes a transition backwards in time to small oscillations around the stable center (though it is possible that the solution approaches the unstable saddle).

[1]  V. Arnold,et al.  Dynamical Systems III: Mathematical Aspects of Classical and Celestial Mechanics , 1989 .

[2]  Richard Haberman,et al.  Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems , 1979 .

[3]  Frank Hadsell,et al.  6. Functions of a Complex Variable , 1995 .

[4]  N. Lebovitz,et al.  Exchange of Stabilities in Autonomous Systems—II. Vertical Bifurcation , 1977 .

[5]  Richard Haberman,et al.  Separatrix crossing: time-invariant potentials with dissipation , 1990 .

[6]  G. J. M. Marée,et al.  Slow Passage Through a Pitchfork Bifurcation , 1996, SIAM J. Appl. Math..

[7]  E. L. Ince.  Ordinary differential equations , 1927 .

[8]  J. Henrard,et al.  The Adiabatic Invariant in Classical Mechanics , 1993 .

[9]  Athanassios S. Fokas,et al.  On the solvability of Painlevé II and IV , 1992 .

[10]  Richard Haberman,et al.  Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture , 1991 .

[11]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[12]  Cary,et al.  Adiabatic-invariant change due to separatrix crossing. , 1986, Physical review. A, General physics.

[13]  A. Neishtadt,et al.  Chaotic advection in a cubic Stokes flow , 1998 .

[14]  J. Cary,et al.  Phase change between separatrix crossings , 1989 .

[15]  A. Neishtadt,et al.  Passage through a separatrix in a resonance problem with a slowly-varying parameter , 1975 .

[16]  A. Neishtadt,et al.  Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow. , 1996, Chaos.

[17]  H. Walther,et al.  Dynamics reported : expositions in dynamical systems : new series , 1992 .

[18]  Alexander Its,et al.  Isomonodromic Deformation Method in the Theory of Painleve Equations , 1986 .

[19]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[20]  N. Lebovitz,et al.  Exchange of Stabilities in Autonomous Systems , 1975 .

[21]  Harold M. Edwards,et al.  Riemann's Zeta Function , 1974 .

[22]  Richard Haberman,et al.  Connection across a Separatrix with Dissipation , 1994 .

[23]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[24]  D. Bump,et al.  On Riemann's zeta function , 1986 .

[25]  J. Conway,et al.  Functions of a Complex Variable , 1964 .

[26]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .