The Tricomi problem on the existence of homoclinic orbits in dissipative systems

Abstract The principles of the proof of the existence of homoclinic orbits in dissipative dynamical systems are described. The application of these principles in the case of a Lorenz system enables new criteria for the existence of homoclinic orbits to be formulated.

[1]  Limiting sets of trajectories of a pendulum-type system , 1956 .

[2]  Gennady A. Leonov,et al.  Lyapunov functions in the attractors dimension theory , 2012 .

[3]  Gennady A. Leonov,et al.  Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors , 1992 .

[4]  Polina S. Landa,et al.  Stochastic and Chaotic Oscillations , 1992 .

[5]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[6]  G. Leonov.  Strange attractors and classical stability theory , 2006 .

[7]  C. Böhm.  Nuovi criteri di esistenza di soluzioni periodiche di una nota equazione differenziale non lineare , 1953 .

[8]  G. Seifert.  On the existence of certain solutions of a nonlinear differential equation , 1952 .

[9]  V. Belykh,et al.  Qualitative investigation of a system of three differential equations in the theory of phase synchronization PMM vol. 39, n≗ 4, 1975, pp. 642-649 , 1975 .

[10]  G. Seifert.  On stability questions for pendulum-type equations , 1956 .

[11]  G. Leonov,et al.  Frequency-Domain Methods for Nonlinear Analysis: Theory and Applications , 1996 .

[12]  P. Bergé,et al.  L'ordre dans le chaos. , 1984 .

[13]  Gennady A. Leonov.  On the global stability of the lorentz system , 1983 .

[14]  S. E. Khaikin,et al.  Theory of Oscillators , 1966 .

[15]  Xinfu Chen.  Lorenz Equations Part I: Existence and Nonexistence of Homoclinic Orbits , 1996 .

[16]  G. Seifert,et al.  On conditions for stability of solutions of pendulum-type equations , 1955 .

[17]  L. Amerio.  Determinazione delle condizioni di stabilità per gli integrali di un'equazione interessante l'elettrotecnica , 1949 .

[18]  Adolf Giger.  Ein Grenzproblem einer technisch wichtigen nichtlinearen Differentialgleichung , 1956 .

[19]  I. N. Baker.  Structures in Dynamics , 1993 .

[20]  G. Seifert.  On certain solutions of a pendulum-type equation , 1953 .

[21]  James A. Yorke,et al.  Preturbulence: A regime observed in a fluid flow model of Lorenz , 1979 .

[22]  S. Hastings,et al.  A shooting approach to the Lorenz equations , 1992, math/9210217.

[23]  William C. Troy,et al.  A Proof That the Lorenz Equations Have a Homoclinic Orbit , 1994 .

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

[25]  W. Hayes,et al.  On the equation for a damped pendulum under constant torque , 1953 .