The paper studies the construction of the Hamiltonian for circuits built from the (α,β) elements of Chua's periodic table. It starts from the Lagrange function, whose existence is limited to Σ-circuits, i.e., circuits built exclusively from elements located on a common Σ-diagonal of the table. We show that the Hamiltonian can also be constructed via the generalized Tellegen's theorem. According to the ideas of predictive modeling, the resulting Hamiltonian is made up exclusively of the constitutive relations of the elements in the circuit. Within the frame of Ostrogradsky's formalism, the simulation scheme of Σ-circuits is designed and examined with the example of a nonlinear Pais-Uhlenbeck oscillator.

Department of Electrical Engineering University of Defence 662 10 Brno Czech Republic

Department of Microelectronics Brno University of Technology 616 00 Brno Czech Republic

Department of Radio Electronics Brno University of Technology 616 00 Brno Czech Republic

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Fuchs A. Nonlinear Dynamics in Complex Systems. Springer; Berlin/Heidelberg, Germany: 2013. DOI

Chua L.O. CNN: A Paradigm for Complexity. Volume 31 World Scientific; Singapore: 1998. (World Scientific Series on Nonlinear Science Series A).

Caravelli F., Traversa F.L., Di Ventra M. Complex dynamics of memristive circuits: Analytical results and universal slow relaxation. Phys. Rev. 2017;E95:022140. doi: 10.1103/PhysRevE.95.022140. PubMed DOI

Di Ventra M., Traversa F.L. Absence of chaos in digital memcomputing machines with solutions. Phys. Lett. A. 2017;381:3255–3257. doi: 10.1016/j.physleta.2017.08.040. PubMed DOI

Chua L.O. Device modeling via nonlinear circuit elements. IEEE Trans. Circuits Syst. 1980;27:1014–1044. doi: 10.1109/TCS.1980.1084742. DOI

Chua L.O. Memristor—The missing circuit element. IEEE Trans. Circuit Theory. 1971;18:507–519. doi: 10.1109/TCT.1971.1083337. DOI

Strukov D.B., Snider G.S., Stewart D.R., Williams R.S. The missing memristor found. Nature. 2008;453:80–83. doi: 10.1038/nature06932. PubMed DOI

Biolek D., Biolek Z., Biolková V. Every nonlinear element from Chua’s table can generate pinched hysteresis loops: Generalised homothety theorem. Electron. Lett. 2016;52:1744–1746. doi: 10.1049/el.2016.2961. DOI

Jeltsema D. Memory Elements: A Paradigm Shift in Lagrangian Modeling of Electrical Circuits. IFAC Proc. Vol. 2012;45:445–450. doi: 10.3182/20120215-3-AT-3016.00078. DOI

Cohen G.Z., Pershin Y.V., Di Ventra M. Lagrange Formalism of Memory Circuit Elements: Classical and Quantum Formulations. Phys. Rev. B. 2012;85:165428–165430. doi: 10.1103/PhysRevB.85.165428. DOI

Biolek Z., Biolek D., Biolková V. Utilization of Euler-Lagrange Equations in Circuits with Memory Elements. Radioengineering. 2016;25:783–789. doi: 10.13164/re.2016.0783. DOI

Biolek Z., Biolek D., Biolková V. Euler-Lagrange Equations of Networks with Higher-Order Elements. Radioengineering. 2017;26:397–405. doi: 10.13164/re.2017.0397. DOI

Millar W. Some general theorems for nonlinear systems possessing resistance. Philos. Mag. 1951;42:1150–1160. doi: 10.1080/14786445108561361. DOI

Cherry E.C. Some general theorems for nonlinear systems possessing reactance. Philos. Mag. 1951;42:1161–1177. doi: 10.1080/14786445108561362. DOI

Pais A., Uhlenbeck G.E. On Field Theories with Non-Localized Action. Phys. Rev. 1950;79:145–165. doi: 10.1103/PhysRev.79.145. DOI

Biolek D., Biolek Z., Biolková V. Lagrangian for Circuits with Higher-Order Elements. Entropy. 2019;21:1059. doi: 10.3390/e21111059. DOI

Ostrogradsky M. Mémoire sur les defférentielles relatives aux problemes des isopérimetres. Mem. Acad. Sci. St. Petersb. 1850;6:385–517.

Lanczos C. The Variational Principles of Mechanics. 4th ed. Dover Publications; Mineola, NY, USA: 1986. Dover Books on Physics (Book 4)

Knetter C.G. Effective Lagrangians with higher derivatives and equations of motion. Phys. Rev. D. 1994;49:6709. doi: 10.1103/PhysRevD.49.6709. PubMed DOI

Deng Y., Li Y. A memristive conservative chaotic circuit consisting of a memristor and capacitor. Chaos. 2020;30:013120. doi: 10.1063/1.5128384. PubMed DOI

Itoh M., Chua L.O. Memristor Hamiltonian Circuits. Int. J. Bif. Chaos. 2011;21:2395–24255. doi: 10.1142/S021812741103012X. DOI

Itoh M., Chua L.O. Dynamics of Hamiltonian Systems and Memristor Circuits. Int. J. Bif. Chaos. 2017;27:1730005. doi: 10.1142/S0218127417300051. DOI

Penfield P., Spence R., Duinker S. A generalized form of Tellegen’s theorem. IEEE Trans. Circuit Theory. 1970;17:302–305. doi: 10.1109/TCT.1970.1083145. DOI

Biolek Z., Biolek D., Biolková V., Kolka Z. Taxicab Geometry in Table of Higher-Order Elements. Nonlinear Dyn. 2019;98:623–636. doi: 10.1007/s11071-019-05218-9. DOI

Woodard R.P. The Theorem of Ostrogradsky. arXiv. 20151506.02210

Soliman A.M. Realizations of ideal FDNC and FDNR elements using new types of mutators. Int. J. Electron. 1978;44:317–323. doi: 10.1080/00207217808900822. DOI

Barbu A., Zhu S.-C. Monte Carlo Methods. 4th ed. Springer; Singapore: 2020. p. 422. DOI

Joglekar Y.N., Wolf S.J. The elusive memristor: Properties of basic electrical circuits. Eur. J. Phys. 2009;30:661–675. doi: 10.1088/0143-0807/30/4/001. DOI

Biolek Z., Biolek D. Memristor and Memristive Systems. Springer; New York, NY, USA: 2014. Fourth Fundamental Circuit Element: SPICE Modeling and Simulation; pp. 105–162. DOI

Lewin L. Dilogarithms and Associated Functions. 1st ed. MacDonald; London, UK: 1958. p. 353.

Biolek D., Biolek Z., Biolková V. Duality of Complex Systems Built from Higher-Order Elements. Complexity. 2018;2018:15. doi: 10.1155/2018/5719397. DOI

Ascoli A., Corinto F., Senger V., Tetzlaff R. Memristor Model Comparison. IEEE Circuits Syst. Mag. 2013;13:89–105. doi: 10.1109/MCAS.2013.2256272. DOI