# American Institute of Mathematical Sciences

August  2022, 27(8): 4573-4588. doi: 10.3934/dcdsb.2021243

## Integrability and bifurcation of a three-dimensional circuit differential system

 1 Faculty of Energy Technology, University of Maribor, Hočevarjev trg 1, 8270 Krško, Slovenia 2 Faculty of Electrical Engineering and Computer Science, University of Maribor, Krško c. 46, 2000 Maribor, Slovenia 3 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia 4 Faculty of Natural Sciences and Mathematics, University of Maribor, Krško c. 160, 2000 Maribor, Slovenia 5 School of Mathematical Sciences, Institute of Modern Analysis-A Frontier Research Center of Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, China 6 Shanghai Engineering Center for Microsatellites, Shanghai, 201203, China

*Corresponding author: Yilei Tang

Received  January 2021 Revised  August 2021 Published  August 2022 Early access  October 2021

We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.

Citation: Brigita Ferčec, Valery G. Romanovski, Yilei Tang, Ling Zhang. Integrability and bifurcation of a three-dimensional circuit differential system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4573-4588. doi: 10.3934/dcdsb.2021243
##### References:
 [1] F. Battelli and M. Fečkan, On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3043-3061.  doi: 10.3934/dcdsb.2017162. [2] S. Belghith, Symbolic dynamics in nondifferentiable system originating in R-L-diode driven circuit, Discrete Contin. Dyn. Syst., 6 (2000), 275-292.  doi: 10.3934/dcds.2000.6.275. [3] S. M. Booker, P. D. Smith, P. Brennan and R. Bullock, In-band disruption of a nonlinear circuit using optimal forcing functions, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 221-242.  doi: 10.3934/dcdsb.2002.2.221. [4] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [5] H. Chen and H. Chen, Global dynamics of a Wilson polynomial Liénard equation, Proc. Amer. Math. Soc., 148 (2020), 4769-4780.  doi: 10.1090/proc/15074. [6] F. Caravelli, Asymptotic behavior of memristive circuits, Entropy, 21 (2019), 19pp. doi: 10.3390/e21080789. [7] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. [8] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639. [9] L. Chua, Memristor the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.  doi: 10.1109/TCT.1971.1083337. [10] I. E. Colak, J. Llibre and C. Valls, Local analytic first integrals of planar analytic differential systems, Phys. Lett. A, 377 (2013), 1065-1069.  doi: 10.1016/j.physleta.2013.03.001. [11] R. Cristiano, T. Carvalho, D. J. Tonon and D. J. Pagano, Hopf and homoclinic bifurcations on the sliding vector field of switching systems in $\mathbb{R}^3$: A case study in power electronics, Phys. D, 347 (2017), 12-20.  doi: 10.1016/j.physd.2017.02.005. [12] W. Cong, J. Llibre and X. Zhang, Generalized rational first integrals of analytic differential systems, J. Differ. Equations, 251 (2011), 2770-2788.  doi: 10.1016/j.jde.2011.05.016. [13] F. D'Annibale, G. Rosi and A. Luongo, On the failure of the 'similar piezoelectric control' in preventing loss of stability by nonconservative positional forces, Z. Angew. Math. Phys., 66 (2015), 1949-1968.  doi: 10.1007/s00033-014-0477-7. [14] Z. Galias and W. Tucker, Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor, J. Differential Equations, 266 (2019), 2408-2434.  doi: 10.1016/j.jde.2018.08.035. [15] I. A. García, J. Llibre and S. Maza, On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations, Nonlinearity, 31 (2018), 2666-2688.  doi: 10.1088/1361-6544/aab592. [16] J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156.  doi: 10.1016/j.jde.2015.11.005. [17] J. Giné and C. Valls, Center problem in the center manifold for quadratic differential systems in $\mathbb{R}^3$, J. Symbolic Comput., 73 (2016), 250-267.  doi: 10.1016/j.jsc.2015.04.001. [18] J. Giné and C. Valls, The generalized polynomial Moon-Rand system, Nonlinear Anal. Real World Appl., 39 (2018), 411-417.  doi: 10.1016/j.nonrwa.2017.07.006. [19] V. F. Edneral, A. Mahdi, V. G. Romanovski and D. S. Shafer, The center problem on a center manifold in $\mathbb{R}^3$, Nonlinear Anal., 75 (2012), 2614-2622.  doi: 10.1016/j.na.2011.11.006. [20] G. A. Leonov, V. I. Vagaitsev and N. V. Kuznetsov, Hidden attractor in smooth Chua systems, Physica D, 241 (2012), 1482-1486.  doi: 10.1016/j.physd.2012.05.016. [21] Y. Li and V. G. Romanovski, Isochronous solutions of a 3-dim symmetric quadratic system, Appl. Math. Comput., 405 (2021), 12pp. doi: 10.1016/j.amc.2021.126250. [22] J. Llibre, D. D. Novaes and C. A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Phys. D, 353/354 (2017), 1-10.  doi: 10.1016/j.physd.2017.05.003. [23] J. Llibre and D. Xiao, Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems, Proc. Amer. Math. Soc., 142 (2014), 2047-2062.  doi: 10.1090/S0002-9939-2014-11923-X. [24] A. Lyapunov, Problème général de la Stabilité du Mouvement, Annals of Mathematics Studies, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947. [25] A. Mahdi, C. Pessoa and J. D. Hauenstein, A hybrid symbolic-numerical approach to the center-focus problem, J. Symbolic Comput., 82 (2017), 57-73.  doi: 10.1016/j.jsc.2016.11.019. [26] B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.  doi: 10.1142/S0218127410027076. [27] H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat.Palermo, 5 (1891), 161–191, Rend. Circ. Mat. Palermo, 11 (1897) 193-239. [28] M. F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations, Amer. J. Math., 103 (1981), 661-682.  doi: 10.2307/2374045. [29] M. F. Singer, Liouvillian first integrals of differential equations, Trans. Am. Math. Soc., 333 (1992), 673-688.  doi: 10.1090/S0002-9947-1992-1062869-X. [30] P. S. Swathy and K. Thamilmaran, Hyperchaos in SC-CNN based modified canonical Chua's circuit, Nonlinear Dynam., 78 (2014), 2639-2650.  doi: 10.1007/s11071-014-1615-7. [31] Z. Wei, I. Moroz, J. C. Sprott, A. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo, Chaos, 27 (2017), 10pp. doi: 10.1063/1.4977417. [32] K. Wu and X. Zhang, Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces, Physica D, 244 (2013), 25-35.  doi: 10.1016/j.physd.2012.10.011. [33] Z.-F. Zhang, T.-R. Ding, W.-Z. Huang and Z.-X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.

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##### References:
 [1] F. Battelli and M. Fečkan, On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3043-3061.  doi: 10.3934/dcdsb.2017162. [2] S. Belghith, Symbolic dynamics in nondifferentiable system originating in R-L-diode driven circuit, Discrete Contin. Dyn. Syst., 6 (2000), 275-292.  doi: 10.3934/dcds.2000.6.275. [3] S. M. Booker, P. D. Smith, P. Brennan and R. Bullock, In-band disruption of a nonlinear circuit using optimal forcing functions, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 221-242.  doi: 10.3934/dcdsb.2002.2.221. [4] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [5] H. Chen and H. Chen, Global dynamics of a Wilson polynomial Liénard equation, Proc. Amer. Math. Soc., 148 (2020), 4769-4780.  doi: 10.1090/proc/15074. [6] F. Caravelli, Asymptotic behavior of memristive circuits, Entropy, 21 (2019), 19pp. doi: 10.3390/e21080789. [7] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. [8] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639. [9] L. Chua, Memristor the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.  doi: 10.1109/TCT.1971.1083337. [10] I. E. Colak, J. Llibre and C. Valls, Local analytic first integrals of planar analytic differential systems, Phys. Lett. A, 377 (2013), 1065-1069.  doi: 10.1016/j.physleta.2013.03.001. [11] R. Cristiano, T. Carvalho, D. J. Tonon and D. J. Pagano, Hopf and homoclinic bifurcations on the sliding vector field of switching systems in $\mathbb{R}^3$: A case study in power electronics, Phys. D, 347 (2017), 12-20.  doi: 10.1016/j.physd.2017.02.005. [12] W. Cong, J. Llibre and X. Zhang, Generalized rational first integrals of analytic differential systems, J. Differ. Equations, 251 (2011), 2770-2788.  doi: 10.1016/j.jde.2011.05.016. [13] F. D'Annibale, G. Rosi and A. Luongo, On the failure of the 'similar piezoelectric control' in preventing loss of stability by nonconservative positional forces, Z. Angew. Math. Phys., 66 (2015), 1949-1968.  doi: 10.1007/s00033-014-0477-7. [14] Z. Galias and W. Tucker, Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor, J. Differential Equations, 266 (2019), 2408-2434.  doi: 10.1016/j.jde.2018.08.035. [15] I. A. García, J. Llibre and S. Maza, On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations, Nonlinearity, 31 (2018), 2666-2688.  doi: 10.1088/1361-6544/aab592. [16] J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156.  doi: 10.1016/j.jde.2015.11.005. [17] J. Giné and C. Valls, Center problem in the center manifold for quadratic differential systems in $\mathbb{R}^3$, J. Symbolic Comput., 73 (2016), 250-267.  doi: 10.1016/j.jsc.2015.04.001. [18] J. Giné and C. Valls, The generalized polynomial Moon-Rand system, Nonlinear Anal. Real World Appl., 39 (2018), 411-417.  doi: 10.1016/j.nonrwa.2017.07.006. [19] V. F. Edneral, A. Mahdi, V. G. Romanovski and D. S. Shafer, The center problem on a center manifold in $\mathbb{R}^3$, Nonlinear Anal., 75 (2012), 2614-2622.  doi: 10.1016/j.na.2011.11.006. [20] G. A. Leonov, V. I. Vagaitsev and N. V. Kuznetsov, Hidden attractor in smooth Chua systems, Physica D, 241 (2012), 1482-1486.  doi: 10.1016/j.physd.2012.05.016. [21] Y. Li and V. G. Romanovski, Isochronous solutions of a 3-dim symmetric quadratic system, Appl. Math. Comput., 405 (2021), 12pp. doi: 10.1016/j.amc.2021.126250. [22] J. Llibre, D. D. Novaes and C. A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Phys. D, 353/354 (2017), 1-10.  doi: 10.1016/j.physd.2017.05.003. [23] J. Llibre and D. Xiao, Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems, Proc. Amer. Math. Soc., 142 (2014), 2047-2062.  doi: 10.1090/S0002-9939-2014-11923-X. [24] A. Lyapunov, Problème général de la Stabilité du Mouvement, Annals of Mathematics Studies, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947. [25] A. Mahdi, C. Pessoa and J. D. Hauenstein, A hybrid symbolic-numerical approach to the center-focus problem, J. Symbolic Comput., 82 (2017), 57-73.  doi: 10.1016/j.jsc.2016.11.019. [26] B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.  doi: 10.1142/S0218127410027076. [27] H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat.Palermo, 5 (1891), 161–191, Rend. Circ. Mat. Palermo, 11 (1897) 193-239. [28] M. F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations, Amer. J. Math., 103 (1981), 661-682.  doi: 10.2307/2374045. [29] M. F. Singer, Liouvillian first integrals of differential equations, Trans. Am. Math. Soc., 333 (1992), 673-688.  doi: 10.1090/S0002-9947-1992-1062869-X. [30] P. S. Swathy and K. Thamilmaran, Hyperchaos in SC-CNN based modified canonical Chua's circuit, Nonlinear Dynam., 78 (2014), 2639-2650.  doi: 10.1007/s11071-014-1615-7. [31] Z. Wei, I. Moroz, J. C. Sprott, A. Akgul and W. Zhang, Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo, Chaos, 27 (2017), 10pp. doi: 10.1063/1.4977417. [32] K. Wu and X. Zhang, Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces, Physica D, 244 (2013), 25-35.  doi: 10.1016/j.physd.2012.10.011. [33] Z.-F. Zhang, T.-R. Ding, W.-Z. Huang and Z.-X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.
Global stability of $O$ as $b_2< b_3$ and $c_2>0$
A stable limit cycle of system (1) appears from Hopf bifurcation when $a = 1, b_1 = 0.3, b_2 = 0.5, b_3 = 0.45, c_1 = 1, c_2 = 0.6$ and $c_3 = 1$. The initial value of the orbit is $(0.1, 0.1, -0.1)$
Two limit cycles of system (22) can be bifurcated from stable weak focus $(0,0)$. Here, the outer limit cycle is stable and the inner limit cycle is unstable when $B_2 = 1, B_3 = -0.189$, $C_1 = 2$ and $h_2 = -1.1$
Orbits of system (22) distribute in the invariant foliations
Intersections of invariant surfaces when $(a, b_1, b_2, b_3, c_1, c_2, c_3) = (1, 1/3, 0, 0.1, 1, 0, 1)$
Eigenvalues of Jacobian Matrix at $O$ and $E_*$
 Conditions eigenvalues at $O$ eigenvalues at $E_*$ $\Delta <0$ $( b_2-b_3\pm \sqrt{-\Delta_0} i)/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2\pm \sqrt{-\Delta_*} i)/2$, $\hat{\lambda}_3$ $\Delta=0$ $( b_2-b_3)/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2)/2$, $\hat{\lambda}_3$ $\Delta >0$ $( b_2-b_3 \pm \sqrt{\Delta_0})/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2\pm \sqrt{\Delta_*} )/2$, $\hat{\lambda}_3$
 Conditions eigenvalues at $O$ eigenvalues at $E_*$ $\Delta <0$ $( b_2-b_3\pm \sqrt{-\Delta_0} i)/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2\pm \sqrt{-\Delta_*} i)/2$, $\hat{\lambda}_3$ $\Delta=0$ $( b_2-b_3)/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2)/2$, $\hat{\lambda}_3$ $\Delta >0$ $( b_2-b_3 \pm \sqrt{\Delta_0})/2$, $\lambda_3$ $( b_2-b_3-b_2z_*^2\pm \sqrt{\Delta_*} )/2$, $\hat{\lambda}_3$
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