Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

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March 2022, 27(3): 1227-1262. doi: 10.3934/dcdsb.2021088

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

Simone Fiori ^1,, , Italo Cervigni ^2, , Mattia Ippoliti ^2, and Claudio Menotta ^2,

1.	Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, 60131, Ancona (Italy)
2.	Corso di Laurea Magistrale in Ingegneria Informatica e dell'Automazione, Università Politecnica delle Marche, 60131, Ancona (Italy)

^* Corresponding author: Simone Fiori

Received July 2020 Revised January 2021 Published March 2022 Early access March 2021

The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.

Keywords: Nonlinear oscillator, Riemannian manifold, Lie group, Second-order (dynamical) system, Nonlinear (active/passive) damping, Numerical calculus on manifold.

Mathematics Subject Classification: 37J15, 37M05, 37N35.

Citation: Simone Fiori, Italo Cervigni, Mattia Ippoliti, Claudio Menotta. Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1227-1262. doi: 10.3934/dcdsb.2021088

References:

[1]	S. Aoi and K. Tsuchiya, Locomotion control of a biped robot using nonlinear oscillators, Autonomous Robots, 19 (2005), 219-232.
[2]	L. Bahar and H. Kwatny, Generalized Lagrangian and conservation law for the damped harmonic oscillator, American Journal of Physics, 49 (1981), 1062-1065. doi: 10.1119/1.12644.
[3]	R. Burston, "Earth-like" planetary magnetotails as non-linear oscillators, Annales Geophysicae Discussions, 2020 (2020), 1-32.
[4]	A. Cammarano, A. Gonzalez-Buelga, S. Neild, D. Wagg, S. Burrow and D. Inman, Optimum load for energy harvesting with non-linear oscillators, in Special Topics in Structural Dynamics - Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013, 6 (2013), 555–560.
[5]	J. Cariñena, J. de Lucas and M. Rañada, Jacobi multipliers, non-local symmetries, and nonlinear oscillators, Journal of Mathematical Physics, 56 (2015), 18pp. doi: 10.1063/1.4922509.
[6]	C. Chen, D. Zanette, D. Czaplewski, S. Shaw and D. López, Direct observation of coherent energy transfer in nonlinear micromechanical oscillators, Nature Communication, 8 (2017), 15523. doi: 10.1038/ncomms15523.
[7]	I. Cornfeld, S. Fomin and Y. Sinai, Smooth Dynamical Systems on Smooth Manifolds, , in Ergodic Theory. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 245, Springer, New York, NY, 1982.
[8]	M. Eie and S.-T. Minking, A Course on Abstract Algebra, World Scientific, 2010. doi: 10.1142/7275.
[9]	K. Engøand and A. Marthinsen, Modeling and solution of some mechanical problems on Lie groups, Multibody System Dynamics, 2 (1998), 71-88.
[10]	S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40. doi: 10.1007/s11424-015-4063-7.
[11]	S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222. doi: 10.1016/j.cnsns.2016.11.025.
[12]	S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100. doi: 10.1007/s11071-018-4546-x.
[13]	M. Fornasier, H. Huang, L. Pareschi and P. Sünnen, Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit, Mathematical Models and Methods in Applied Sciences, 30 (2020), 2725-2751. doi: 10.1142/S0218202520500530.
[14]	S. Gajbhiye and R. Banavar, The Euler-Poincaré equations for a spherical robot actuated by a pendulum, IFAC Proceedings Volumes, 45 (2012), 72-77. doi: 10.3182/20120829-3-IT-4022.00011.
[15]	H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley, 1980.
[16]	H. Goto, K. Tatsumura and A. Dixon, Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems, Science Advances, 5 (2019), eaav2372. doi: 10.1126/sciadv.aav2372.
[17]	F. Hajdu, Numerical examination of nonlinear oscillators, Pollack Periodica, 13 (2018), 95-106.
[18]	J.-H. He, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 1252-1260.
[19]	M. Holmes, Conservative numerical methods for nonlinear oscillators, American Journal of Physics, 88 (2020), 60-69. doi: 10.1119/10.0000295.
[20]	A. Iserles, Numerical methods on (and off) manifolds, in Foundations of Computational Mathematics (ed. S. M. E. Cucker F.), Springer, Berlin, Heidelberg, 1997,180–189.
[21]	N. Khan, K. Nasir Uddin and K. Nadeem Alam, Accurate numerical solutions of conservative nonlinear oscillators, Nonlinear Engineering, 3 (2014), 197-201. doi: 10.1515/nleng-2014-0009.
[22]	I. Kovacic, Conservation laws of two coupled non-linear oscillators, International Journal of Non-Linear Mechanics, 41 (2006), 751–760, https://eprints.soton.ac.uk/43513/. doi: 10.1016/j.ijnonlinmec.2006.04.007.
[23]	I. Kovacic, Four types of strongly nonlinear oscillators: Generalization of a perturbation procedure, Procedia IUTAM, 19 (2016), 101–109, IUTAM Symposium Analytical Methods in Nonlinear Dynamics.
[24]	I. Kovacic and M. Brennan, Background: On Georg Duffing and the Duffing equation, in The Duffing Equation: Nonlinear Oscillators and their Behaviour (eds. I. Kovacic and M. Brennan), John Wiley & Sons, 2011, 1–23. doi: 10.1002/9780470977859.ch1.
[25]	I. Kovacic and M. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977859.
[26]	Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in Araki H. (eds) International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, Springer, Berlin, Heidelberg, 1975,420–422.
[27]	M. Lakshmanan and K. Murali, Harnessing chaos: Synchronization and secure signal transmission, Current Science, 67 (1994), 989-995.
[28]	K. Lee and K. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics, 404 (2020), 108973, 32pp. doi: 10.1016/j.jcp.2019.108973.
[29]	T.-C. Lim, Two-body relationship between the Pearson-Takai-Halicioglu-Tiller and the Biswas-Hamann potential functions, Brazilian Journal of Physics, 35 (2005), 641-644. doi: 10.1590/S0103-97332005000400010.
[30]	A. Lotka, Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences of the United States of America, 6 (1920), 410-415. doi: 10.1073/pnas.6.7.410.
[31]	J. Lu and Y. Liang, Analytical approach to the nonlinear free vibration of a conservative oscillator, Journal of Low Frequency Noise, Vibration and Active Control.
[32]	R. Mickens, Construction of finite difference schemes for coupled nonlinear oscillators derived from a discrete energy function, Journal of Difference Equations and Applications, 2 (1996), 185-193. doi: 10.1080/10236199608808053.
[33]	M. Molaei, Hyperbolic dynamics of discrete dynamical systems on pseudo-Riemannian manifolds, Electronic Research Announcements, 25 (2018), 8-15. doi: 10.3934/era.2018.25.002.
[34]	F. Molero, M. Lara, S. Ferrer and F. Céspedes, 2-D Duffing oscillator: Elliptic functions from a dynamical systems point of view, Qualitative Theory of Dynamical Systems, 12 (2013), 115-139. doi: 10.1007/s12346-012-0081-1.
[35]	D. Moore and E. Spiegel, A thermally excited non-linear oscillator, Astrophysical Journal, 143 (1966), 871-887. doi: 10.1086/148562.
[36]	O. Mustafa, $n$-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance, Physica Scripta, 95.
[37]	I. N'Doye and T. Kirati, Stability and trajectories analysis of a fractional generalization of simple pendulum dynamic equation, in 18th European Control Conference (ECC), 2019, 3854–3860.
[38]	V. Nekorkin, Introduction to Nonlinear Oscillations, Wiley-VCH Verlag GmbH & Co. KGaA, 2015. doi: 10.1002/9783527695942.
[39]	M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
[40]	A. Ouannas, A. Karouma, G. Grassi, V.-T. Pham and V. Luong, A novel secure communications scheme based on chaotic modulation, recursive encryption and chaotic masking, Alexandria Engineering Journal, 60 (2021), 1873-1884.
[41]	A. Prykarpatsky and I. Mykytiuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Mathematics and its Applications, 443. Kluwer Academic Publishers Group, Dordrecht, 1998. doi: 10.1007/978-94-011-4994-5.
[42]	R. Reid, Local phenomenological nucleon-nucleon potentials, Annals of Physics, 50 (1968), 411-448. doi: 10.1016/0003-4916(68)90126-7.
[43]	N. Sherif and E. Morsy, Computing real logarithm of a real matrix, International Journal of Algebra, 2 (2008), 131-142.
[44]	H. Sussmann, Dynamical systems on manifolds: Accessibility and controllability, in 1971 IEEE Conference on Decision and Control, 1971,188–191.
[45]	A. Tero, R. Kobayashi and T. Nakagaki, A coupled-oscillator model with a conservation law for the rhythmic amoeboid movements of plasmodial slime molds, Physica D: Nonlinear Phenomena, 205 (2005), 125-135.
[46]	A. Vakakis, M. King and A. Pearlstein, Forced localization in a periodic chain of non-linear oscillators, International Journal of Non-Linear Mechanics, 29 (1994), 429-447.
[47]	V. Valimaki, J. Nam, J. Smith and J. Abel, Alias-suppressed oscillators based on differentiated polynomial waveforms, IEEE Transactions on Audio, Speech, and Language Processing, 18 (2010), 786-798.
[48]	B. van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051-1086.
[49]	S. Venturini, Continuous dynamical systems on Taut complex manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 291–298, http://www.numdam.org/item/ASNSP_1997_4_24_2_291_0.
[50]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Memoria della Reale Accademia Nazionale dei Lincei, 2 (1926), 31-113.
[51]	G. Wang, X. Chen and S.-K. Han, Central pattern generator and feedforward neural network-based self-adaptive gait control for a crab-like robot locomoting on complex terrain under two reflex mechanisms, International Journal of Advanced Robotic Systems, 14.
[52]	N. Wilkinson, T. Bossomaier, M. Harre and A. Snyder, Strategic planning in the game of Go using coupled non-linear oscillators, in European Conference on Artificial Intelligence (ECAI 2010), IOS Press, 2010, 1095–1096.
[53]	X. Xia and S. Li, Research on improved chaotic particle optimization algorithm based on complex function, Frontiers in Physics, 8 (2020), 368. doi: 10.3389/fphy.2020.00368.
[54]	S. Yu, J. Lü, W. Tang and G. Chen, A general multiscroll Lorenz system family and its realization via digital signal processors, Chaos, 16 (2006), 033126. doi: 10.1063/1.2336739.
[55]	M. Zarei, A. Kalhor and M. Masouleh, An experimental oscillation damping impedance control for the Novint Falcon haptic device based on the phase trajectory length function concept, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233 (2019), 2663-2672. doi: 10.1177/0954406218799779.
[56]	Z. Zhang and G. Chen, Liquid mixing enhancement by chaotic perturbations in stirred tanks, Chaos, Solitons & Fractals, 36 (2008), 144–149, http://www.sciencedirect.com/science/article/pii/S0960077906005947. doi: 10.1016/j.chaos.2006.06.024.
[57]	S. Zhen and G. Davies, Calculation of the Lennard-Jones n-m potential energy parameters for metals, Physica Status Solidi (a), 78 (1983), 595-605.

show all references

References:

[1]	S. Aoi and K. Tsuchiya, Locomotion control of a biped robot using nonlinear oscillators, Autonomous Robots, 19 (2005), 219-232.
[2]	L. Bahar and H. Kwatny, Generalized Lagrangian and conservation law for the damped harmonic oscillator, American Journal of Physics, 49 (1981), 1062-1065. doi: 10.1119/1.12644.
[3]	R. Burston, "Earth-like" planetary magnetotails as non-linear oscillators, Annales Geophysicae Discussions, 2020 (2020), 1-32.
[4]	A. Cammarano, A. Gonzalez-Buelga, S. Neild, D. Wagg, S. Burrow and D. Inman, Optimum load for energy harvesting with non-linear oscillators, in Special Topics in Structural Dynamics - Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013, 6 (2013), 555–560.
[5]	J. Cariñena, J. de Lucas and M. Rañada, Jacobi multipliers, non-local symmetries, and nonlinear oscillators, Journal of Mathematical Physics, 56 (2015), 18pp. doi: 10.1063/1.4922509.
[6]	C. Chen, D. Zanette, D. Czaplewski, S. Shaw and D. López, Direct observation of coherent energy transfer in nonlinear micromechanical oscillators, Nature Communication, 8 (2017), 15523. doi: 10.1038/ncomms15523.
[7]	I. Cornfeld, S. Fomin and Y. Sinai, Smooth Dynamical Systems on Smooth Manifolds, , in Ergodic Theory. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 245, Springer, New York, NY, 1982.
[8]	M. Eie and S.-T. Minking, A Course on Abstract Algebra, World Scientific, 2010. doi: 10.1142/7275.
[9]	K. Engøand and A. Marthinsen, Modeling and solution of some mechanical problems on Lie groups, Multibody System Dynamics, 2 (1998), 71-88.
[10]	S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40. doi: 10.1007/s11424-015-4063-7.
[11]	S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222. doi: 10.1016/j.cnsns.2016.11.025.
[12]	S. Fiori, Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications, Nonlinear Dynamics, 94 (2018), 3077-3100. doi: 10.1007/s11071-018-4546-x.
[13]	M. Fornasier, H. Huang, L. Pareschi and P. Sünnen, Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit, Mathematical Models and Methods in Applied Sciences, 30 (2020), 2725-2751. doi: 10.1142/S0218202520500530.
[14]	S. Gajbhiye and R. Banavar, The Euler-Poincaré equations for a spherical robot actuated by a pendulum, IFAC Proceedings Volumes, 45 (2012), 72-77. doi: 10.3182/20120829-3-IT-4022.00011.
[15]	H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley, 1980.
[16]	H. Goto, K. Tatsumura and A. Dixon, Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems, Science Advances, 5 (2019), eaav2372. doi: 10.1126/sciadv.aav2372.
[17]	F. Hajdu, Numerical examination of nonlinear oscillators, Pollack Periodica, 13 (2018), 95-106.
[18]	J.-H. He, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 1252-1260.
[19]	M. Holmes, Conservative numerical methods for nonlinear oscillators, American Journal of Physics, 88 (2020), 60-69. doi: 10.1119/10.0000295.
[20]	A. Iserles, Numerical methods on (and off) manifolds, in Foundations of Computational Mathematics (ed. S. M. E. Cucker F.), Springer, Berlin, Heidelberg, 1997,180–189.
[21]	N. Khan, K. Nasir Uddin and K. Nadeem Alam, Accurate numerical solutions of conservative nonlinear oscillators, Nonlinear Engineering, 3 (2014), 197-201. doi: 10.1515/nleng-2014-0009.
[22]	I. Kovacic, Conservation laws of two coupled non-linear oscillators, International Journal of Non-Linear Mechanics, 41 (2006), 751–760, https://eprints.soton.ac.uk/43513/. doi: 10.1016/j.ijnonlinmec.2006.04.007.
[23]	I. Kovacic, Four types of strongly nonlinear oscillators: Generalization of a perturbation procedure, Procedia IUTAM, 19 (2016), 101–109, IUTAM Symposium Analytical Methods in Nonlinear Dynamics.
[24]	I. Kovacic and M. Brennan, Background: On Georg Duffing and the Duffing equation, in The Duffing Equation: Nonlinear Oscillators and their Behaviour (eds. I. Kovacic and M. Brennan), John Wiley & Sons, 2011, 1–23. doi: 10.1002/9780470977859.ch1.
[25]	I. Kovacic and M. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley & Sons, Ltd., Chichester, 2011. doi: 10.1002/9780470977859.
[26]	Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in Araki H. (eds) International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, Springer, Berlin, Heidelberg, 1975,420–422.
[27]	M. Lakshmanan and K. Murali, Harnessing chaos: Synchronization and secure signal transmission, Current Science, 67 (1994), 989-995.
[28]	K. Lee and K. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics, 404 (2020), 108973, 32pp. doi: 10.1016/j.jcp.2019.108973.
[29]	T.-C. Lim, Two-body relationship between the Pearson-Takai-Halicioglu-Tiller and the Biswas-Hamann potential functions, Brazilian Journal of Physics, 35 (2005), 641-644. doi: 10.1590/S0103-97332005000400010.
[30]	A. Lotka, Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences of the United States of America, 6 (1920), 410-415. doi: 10.1073/pnas.6.7.410.
[31]	J. Lu and Y. Liang, Analytical approach to the nonlinear free vibration of a conservative oscillator, Journal of Low Frequency Noise, Vibration and Active Control.
[32]	R. Mickens, Construction of finite difference schemes for coupled nonlinear oscillators derived from a discrete energy function, Journal of Difference Equations and Applications, 2 (1996), 185-193. doi: 10.1080/10236199608808053.
[33]	M. Molaei, Hyperbolic dynamics of discrete dynamical systems on pseudo-Riemannian manifolds, Electronic Research Announcements, 25 (2018), 8-15. doi: 10.3934/era.2018.25.002.
[34]	F. Molero, M. Lara, S. Ferrer and F. Céspedes, 2-D Duffing oscillator: Elliptic functions from a dynamical systems point of view, Qualitative Theory of Dynamical Systems, 12 (2013), 115-139. doi: 10.1007/s12346-012-0081-1.
[35]	D. Moore and E. Spiegel, A thermally excited non-linear oscillator, Astrophysical Journal, 143 (1966), 871-887. doi: 10.1086/148562.
[36]	O. Mustafa, $n$-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance, Physica Scripta, 95.
[37]	I. N'Doye and T. Kirati, Stability and trajectories analysis of a fractional generalization of simple pendulum dynamic equation, in 18th European Control Conference (ECC), 2019, 3854–3860.
[38]	V. Nekorkin, Introduction to Nonlinear Oscillations, Wiley-VCH Verlag GmbH & Co. KGaA, 2015. doi: 10.1002/9783527695942.
[39]	M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
[40]	A. Ouannas, A. Karouma, G. Grassi, V.-T. Pham and V. Luong, A novel secure communications scheme based on chaotic modulation, recursive encryption and chaotic masking, Alexandria Engineering Journal, 60 (2021), 1873-1884.
[41]	A. Prykarpatsky and I. Mykytiuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Mathematics and its Applications, 443. Kluwer Academic Publishers Group, Dordrecht, 1998. doi: 10.1007/978-94-011-4994-5.
[42]	R. Reid, Local phenomenological nucleon-nucleon potentials, Annals of Physics, 50 (1968), 411-448. doi: 10.1016/0003-4916(68)90126-7.
[43]	N. Sherif and E. Morsy, Computing real logarithm of a real matrix, International Journal of Algebra, 2 (2008), 131-142.
[44]	H. Sussmann, Dynamical systems on manifolds: Accessibility and controllability, in 1971 IEEE Conference on Decision and Control, 1971,188–191.
[45]	A. Tero, R. Kobayashi and T. Nakagaki, A coupled-oscillator model with a conservation law for the rhythmic amoeboid movements of plasmodial slime molds, Physica D: Nonlinear Phenomena, 205 (2005), 125-135.
[46]	A. Vakakis, M. King and A. Pearlstein, Forced localization in a periodic chain of non-linear oscillators, International Journal of Non-Linear Mechanics, 29 (1994), 429-447.
[47]	V. Valimaki, J. Nam, J. Smith and J. Abel, Alias-suppressed oscillators based on differentiated polynomial waveforms, IEEE Transactions on Audio, Speech, and Language Processing, 18 (2010), 786-798.
[48]	B. van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051-1086.
[49]	S. Venturini, Continuous dynamical systems on Taut complex manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 291–298, http://www.numdam.org/item/ASNSP_1997_4_24_2_291_0.
[50]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Memoria della Reale Accademia Nazionale dei Lincei, 2 (1926), 31-113.
[51]	G. Wang, X. Chen and S.-K. Han, Central pattern generator and feedforward neural network-based self-adaptive gait control for a crab-like robot locomoting on complex terrain under two reflex mechanisms, International Journal of Advanced Robotic Systems, 14.
[52]	N. Wilkinson, T. Bossomaier, M. Harre and A. Snyder, Strategic planning in the game of Go using coupled non-linear oscillators, in European Conference on Artificial Intelligence (ECAI 2010), IOS Press, 2010, 1095–1096.
[53]	X. Xia and S. Li, Research on improved chaotic particle optimization algorithm based on complex function, Frontiers in Physics, 8 (2020), 368. doi: 10.3389/fphy.2020.00368.
[54]	S. Yu, J. Lü, W. Tang and G. Chen, A general multiscroll Lorenz system family and its realization via digital signal processors, Chaos, 16 (2006), 033126. doi: 10.1063/1.2336739.
[55]	M. Zarei, A. Kalhor and M. Masouleh, An experimental oscillation damping impedance control for the Novint Falcon haptic device based on the phase trajectory length function concept, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233 (2019), 2663-2672. doi: 10.1177/0954406218799779.
[56]	Z. Zhang and G. Chen, Liquid mixing enhancement by chaotic perturbations in stirred tanks, Chaos, Solitons & Fractals, 36 (2008), 144–149, http://www.sciencedirect.com/science/article/pii/S0960077906005947. doi: 10.1016/j.chaos.2006.06.024.
[57]	S. Zhen and G. Davies, Calculation of the Lennard-Jones n-m potential energy parameters for metals, Physica Status Solidi (a), 78 (1983), 595-605.

Figure 1. Example of Duffing potentials corresponding to different combinations of signs and different values of $ \kappa $. The $ (++) $ combination is referred to as hard Duffing potential while the combination $ (+-) $ is referred to as soft Duffing potential. (A combination $ (-+) $, not shown in this figure, is referred to as double-well potential.)

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Figure 2. Example of Keplerian potential with critical distance $ d_\mathrm{c} = 1 $

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Figure 3. Behavior of the simple pendulum (46) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.2\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.001 $

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Figure 4. Behavior of the simple pendulum (46) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.8\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

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Figure 5. Behavior of the hard Duffing oscillator (48) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.4\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0005 $

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Figure 6. Behavior of the hard Duffing oscillator (48) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.001 $

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Figure 7. Behavior of the soft Duffing oscillator (49) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0001 $

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Figure 8. Behavior of the soft Duffing oscillator (49) in the presence of non-linear damping, implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.2 $, $ \epsilon = 1.3 $ and $ h = 0.0001 $

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Figure 9. Behavior of the soft Duffing oscillator (50) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Heun method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0001 $

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Figure 10. Behavior of the soft Duffing oscillator (51) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Runge-Kutta method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ h = 0.0001 $

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Figure 11. Behavior of the double-well Duffing oscillator (52) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.8 $ and $ h = 0.0001 $

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Figure 12. Behavior of the double-well Duffing oscillator (52) in the presence of non-linear damping, implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 1.4 $, $ \mu = 0.2 $, $ \epsilon = 1.3 $ and $ h = 0.0001 $

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Figure 13. Behavior of the Van der Pol oscillator (53) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.2\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.001 $

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Figure 14. Behavior of Van der Pol oscillator (53) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

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Figure 15. Behavior of the Kepler oscillator (54) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.3\ 0]^\top $. The values of the parameters used in this simulation are $ \lambda = 0.5 $, $ \rho = 0.08 $ and $ h = 0.001 $ (hence, the critical distance is $ d_\mathrm{c} = 0.4 $)

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Figure 16. Behavior of the Kepler oscillator (54) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \rho = 0.08 $, $ \mu = 0.1 $, $ \epsilon = 2 $ and $ \lambda = 0.5 $ and $ h = 0.002 $ (hence, the critical distance is $ d_\mathrm{c} = 0.4 $)

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Figure 17. Behavior of the classic pendulum oscillator (58) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.002 $

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Figure 18. Behavior of the simple pendulum (58) in the presence of non-linear damping. The left-hand panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.3 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

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Figure 19. Behavior of a hard Duffing oscillator (59) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0005 $

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Figure 20. Behavior of a hard Duffing oscillator (59) in the presence of non-linear damping. The left-hand panel shows the trajectory in the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the refernce point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.0008 $

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Figure 21. Behavior of the Van der Pol oscillator (60) on the special orthogonal group $ SO(3) $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the initial point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 1 $ and $ h = 0.0005 $

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Figure 22. Behavior of the Van der Pol (60) in the presence of non-linear damping. The left-hand panel shows the trajectory in the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the initial point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 1 $, $ \mu = 0.5 $, $ \epsilon = 3 $ and $ h = 0.001 $

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Figure 23. Behavior of the Kepler oscillator (61) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \lambda = 0.2 $, $ \rho = 0.8 $ and $ h = 0.0003 $

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Figure 24. Behavior of the Kepler oscillator (61) in the presence of non-linear damping. The left-hand panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \lambda = 0.5 $, $ \mu = 0.3 $, $ \rho = 0.1 $, $ \epsilon = 1.5 $ and $ h = 0.0003 $

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Figure 25. Behavior of the classic pendulum oscillator (31) on two different manifolds, in the absence of non-linear damping (namely, $ \mu = 0 $). Indicating by $ k $ the discrete-time index, the figure (a) shows the configuration of the system at $ k = 2,301 $, the figure (b) at $ k = 4,461 $, the figure (c) at $ k = 9,321 $ and the figure (d) shows where the simulations stops, at $ k = 15,000 $

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Figure 26. Behavior of the classic pendulum oscillator (31) on two different manifolds, in the presence of non-linear damping. Indicating by $ k $ the discrete-time stamp, the figure (a) shows the configuration of the system at $ k = 2,641 $, the figure (b) at $ k = 6,381 $, the figure (c) at $ k = 10,381 $ and the figure (d) shows where the simulations stops, at $ k = 15,000 $

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