Higher-Order Hamiltonian for Circuits with (α,β) Elements.

Area of Science:

• Circuit theory
• Nonlinear dynamics
• Mathematical physics

Background:

• Chua's periodic table classifies circuit elements based on their (α,β) properties.
• Hamiltonian construction is crucial for analyzing circuit dynamics, but traditionally limited to specific circuit types (Σ-circuits).

Purpose of the Study:

• To develop a generalized method for constructing the Hamiltonian for circuits built from (α,β) elements of Chua's periodic table.
• To demonstrate that this construction is possible beyond the limitations of Σ-circuits.

Main Methods:

• Derivation starting from the Lagrange function for Σ-circuits.
• Application of the generalized Tellegen's theorem for broader circuit applicability.
• Utilizing Ostrogradsky's formalism for simulation scheme design.

Main Results:

• The Hamiltonian can be constructed using the generalized Tellegen's theorem, extending beyond Σ-circuits.
• The resulting Hamiltonian exclusively comprises the constitutive relations of the circuit elements.
• A simulation scheme for Σ-circuits was designed and validated using a nonlinear Pais-Uhlenbeck oscillator.

Conclusions:

• A novel and generalized approach to Hamiltonian construction for circuits based on Chua's elements is established.
• The method offers a predictive modeling framework, relying solely on element properties.
• The developed simulation scheme provides a tool for analyzing complex nonlinear circuits.