On a Countable Sequence of Homoclinic Orbits Arising Near a Saddle-Center Point

Area of Science:

• Mathematical physics
• Dynamical systems theory
• Nonlinear partial differential equations

Background:

• Singular perturbation theory describes systems with small parameters, often leading to complex behaviors like oscillations.
• Homoclinic orbits are crucial in understanding the global dynamics of systems, particularly near equilibrium points.
• Previous research conjectured that oscillations near saddle-center points might vanish under specific conditions related to complex singularities.

Purpose of the Study:

• To rigorously prove the conjecture regarding the vanishing of oscillations in a specific dynamical system.
• To investigate the conditions under which a true homoclinic orbit exists in the context of singular perturbation.
• To establish a connection between complex analytic extensions of orbits and the existence of vanishing oscillations.

Main Methods:

• Utilizing exponential small splitting of separatrices within singular perturbation theory.
• Analyzing a specific fourth-order nonlinear equation derived from the modified Korteweg-de Vries equation.
• Employing complex analytic extension of the limiting homoclinic orbit to identify singularities.

Main Results:

• A rigorous proof is provided for the conjecture that oscillations can vanish at a countable set of parameter values.
• The existence of a quadruplet of singularities in the complex analytic extension is shown to be a key condition for vanishing oscillations.
• The study confirms the nonexistence of true homoclinic orbits in the general case due to nonvanishing oscillations.

Conclusions:

• The findings confirm a long-standing conjecture in singular perturbation theory concerning the vanishing of oscillations.
• The work provides a rigorous mathematical framework for understanding homoclinic phenomena in nonlinear systems.
• The results have implications for the analysis of traveling wave solutions in models like the modified Korteweg-de Vries equation.