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

Zdeněk Biolek1,2, Dalibor Biolek1,2, Viera Biolková3

  • 1Department of Microelectronics, Brno University of Technology, 616 00 Brno, Czech Republic.

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Summary

This study presents a new method for constructing Hamiltonians in circuits using Chua

Keywords:
Chua’s tableEuler-Lagrange equationHamiltonianLagrangianconstitutive relationhigher-order elementmemristor

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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.