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High-order variational calculation for the frequency of time-periodic solutions
Axel Pelster1, Hagen Kleinert, Michael Schanz
1Institute of Theoretical Physics, Free University of Berlin, Arnimallee 14, D-14195 Berlin, Germany. pelster@physik.fu-berlin.de
Summary
We introduce a new convergent variational perturbation theory to calculate the frequencies of nonlinear dynamical systems. This method is demonstrated effectively on the Duffing oscillator, showcasing its practical application.
Area of Science:
- Nonlinear dynamics
- Theoretical physics
- Applied mathematics
Background:
- Nonlinear dynamical systems often exhibit complex time-periodic solutions.
- Calculating the precise frequencies of these solutions can be computationally challenging.
- Existing perturbation methods may face convergence issues.
Purpose of the Study:
- To develop a novel convergent variational perturbation theory.
- To provide an accurate method for determining the frequency of time-periodic solutions.
- To demonstrate the theory's efficacy using a standard nonlinear system.
Main Methods:
- Formulation of a convergent variational perturbation approach.
- Application of the theory to the Duffing oscillator model.
- Analysis of the frequency of time-periodic solutions.
Main Results:
- The developed theory provides a convergent analytical solution.
- Accurate frequency prediction for the Duffing oscillator was achieved.
- The method overcomes limitations of traditional perturbation techniques.
Conclusions:
- The convergent variational perturbation theory is a powerful tool for analyzing nonlinear systems.
- This approach offers a reliable method for frequency calculation.
- The Duffing oscillator serves as a successful validation case.