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Melnikov-Arnold integrals and optimal normal forms
1Mathematics and Mechanics Department, Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 Saint Petersburg, Russia and Institute of Applied Astronomy, Russian Academy of Sciences, 191187 Saint Petersburg, Russia.
Chaos (Woodbury, N.Y.)
|April 14, 2026
Summary
Melnikov-Arnold integrals (MA-integrals), traditionally used for separatrix splitting, can now estimate secondary resonance sizes in Hamiltonian systems. This new method simplifies analysis of complex resonance structures.
Area of Science:
- Dynamical Systems
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Melnikov-Arnold integrals (MA-integrals) are established tools for analyzing separatrix splitting in Hamiltonian systems.
- Secondary resonances are critical features in the phase space of dynamical systems, influencing long-term behavior.
- Traditional methods for estimating resonance sizes can be computationally intensive and complex.
Purpose of the Study:
- To demonstrate the utility of MA-integrals for estimating the sizes of secondary resonances.
- To introduce a novel, simplified procedure for secondary resonance analysis.
- To validate the MA-integral-based method within the standard map model.
Main Methods:
- Calculation of Melnikov-Arnold integrals (MA-integrals).
- Application of a newly developed MA-based procedure.
- Analysis within the standard map model framework.
- Estimation of secondary resonance sizes for various orders.
Main Results:
- The MA-integral-based procedure effectively estimates secondary resonance sizes.
- The method allows for the estimation of secondary resonances of any order.
- This approach bypasses the need for traditional, cumbersome normalization procedures.
- Successful application demonstrated within the standard map model.
Conclusions:
- MA-integrals offer a powerful and efficient alternative for estimating secondary resonance sizes.
- The developed MA-based procedure simplifies the analysis of complex resonance structures in Hamiltonian systems.
- This method provides a valuable tool for understanding the dynamics of systems like the standard map.
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