doi: 10.3934/dcdsb.2021243
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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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021243
References:
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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.  Google Scholar

[2]

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S. M. BookerP. D. SmithP. 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.  Google Scholar

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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.  Google Scholar

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F. Caravelli, Asymptotic behavior of memristive circuits, Entropy, 21 (2019), 19pp. doi: 10.3390/e21080789.  Google Scholar

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L. Chua, Memristor the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.  doi: 10.1109/TCT.1971.1083337.  Google Scholar

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I. E. ColakJ. 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.  Google Scholar

[11]

R. CristianoT. CarvalhoD. 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.  Google Scholar

[12]

W. CongJ. 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.  Google Scholar

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F. D'AnnibaleG. 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.  Google Scholar

[14]

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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.  Google Scholar

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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.  Google Scholar

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J. LlibreD. 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.  Google Scholar

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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.  Google Scholar

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A. Lyapunov, Problème général de la Stabilité du Mouvement, Annals of Mathematics Studies, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947.  Google Scholar

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A. MahdiC. 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.  Google Scholar

[26]

B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.  doi: 10.1142/S0218127410027076.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z.-F. Zhang, T.-R. Ding, W.-Z. Huang and Z.-X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

S. M. BookerP. D. SmithP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. Caravelli, Asymptotic behavior of memristive circuits, Entropy, 21 (2019), 19pp. doi: 10.3390/e21080789.  Google Scholar

[7] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.   Google Scholar
[8] S. N. ChowC. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.  Google Scholar
[9]

L. Chua, Memristor the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507-519.  doi: 10.1109/TCT.1971.1083337.  Google Scholar

[10]

I. E. ColakJ. 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.  Google Scholar

[11]

R. CristianoT. CarvalhoD. 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.  Google Scholar

[12]

W. CongJ. 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.  Google Scholar

[13]

F. D'AnnibaleG. 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.  Google Scholar

[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.  Google Scholar

[15]

I. A. GarcíaJ. 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.  Google Scholar

[16]

J. GinéJ. LlibreK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. F. EdneralA. MahdiV. 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.  Google Scholar

[20]

G. A. LeonovV. 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.  Google Scholar

[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.  Google Scholar

[22]

J. LlibreD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

A. MahdiC. 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.  Google Scholar

[26]

B. Muthuswamy and L. O. Chua, Simplest chaotic circuit, Internat. J. Bifur. Chaos, 20 (2010), 1567-1580.  doi: 10.1142/S0218127410027076.  Google Scholar

[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. Google Scholar

[28]

M. F. Singer, Liouvillian solutions of nth order homogeneous linear differential equations, Amer. J. Math., 103 (1981), 661-682.  doi: 10.2307/2374045.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z.-F. Zhang, T.-R. Ding, W.-Z. Huang and Z.-X. Dong, Qualitative Theory of Differential Equations, Amer. Math. Soc., Providence, 1992.  Google Scholar

Figure 1.  Global stability of $ O $ as $ b_2< b_3 $ and $ c_2>0 $
Figure 2.  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) $
Figure 3.  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 $
Figure 4.  Orbits of system (22) distribute in the invariant foliations
Figure 5.  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) $
Table 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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