October  2017, 22(8): 3043-3061. doi: 10.3934/dcdsb.2017162

On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits

1. 

Department of Industrial Engeneering and Mathematics, Marche Polytecnic University, Ancona, Italy

2. 

Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia

3. 

Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia

1This paper has been performed within the activity of GNAMPA-INdAM
2Partially supported by the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17 and the Slovak Research and Development Agency under the contract No. APVV-14-0378

Received  July 2016 Revised  November 2016 Published  June 2017

Higher-dimensional nonlinear and perturbed systems of implicit ordinary differential equations are studied by means of methods of dynamical systems. Namely, the persistence of solutions are studied under nonautonomous perturbations connecting either impasse points with IK-singularities or two impasse points. Important parts of the paper are applications of the theory to concrete perturbed fully nonlinear RLC circuits.

Citation: Flaviano Battelli, Michal Fečkan. On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3043-3061. doi: 10.3934/dcdsb.2017162
References:
[1]

F. Battelli and M. Fečkan, Nonlinear RLC circuits and implicit ODEs, Differential Integral Equations, 27 (2014), 671-690. Google Scholar

[2]

F. Battelli and M. Fečkan, Melnikov theory for nonlinear implicit ODEs, J. Differential Equations, 256 (2014), 1157-1190. doi: 10.1016/j.jde.2013.10.012. Google Scholar

[3]

F. Battelli and M. Fečkan, Melnikov theory for weakly coupled nonlinear RLC circuits Bound. Value Probl. , 2014: 101 (2014), 27pp. doi: 10.1186/1687-2770-2014-101. Google Scholar

[4]

F. Battelli and M. Fečkan, On the existence of solutions connecting singularities in nonlinear RLC circuits, Nonlinear Anal., 116 (2015), 26-36. doi: 10.1016/j.na.2014.12.015. Google Scholar

[5]

L. O. Chua, Ch. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits McGraw-Hill, New York, 1987.Google Scholar

[6]

E. Gluskin, A nonlinear resistor and nonlinear inductor using a nonlinear capacitor, J. Franklin Inst., 336 (1999), 1035-1047. doi: 10.1016/S0016-0032(99)00029-0. Google Scholar

[7]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution European Math. Soc. 2006. doi: 10.4171/017. Google Scholar

[8]

N. Lazarides, M. Eleftheriou and G. P. Tsironis, Discrete breathers in nonlinear magnetic metamaterials Phys. Rew. Lett. , 97 (2006), 157406. doi: 10.1103/PhysRevLett. 97. 157406. Google Scholar

[9]

M. Medved', Normal forms of implicit and observed implicit differential equations, Riv. Mat. Pura ed Appl., 10 (1991), 95-107. Google Scholar

[10]

M. Medved', Qualitative properties of generalized vector fields, Riv. Mat. Pura ed Appl., 15 (1994), 7-31. Google Scholar

[11]

P. J. Rabier and W. C. Rheinboldt, A general existence and uniqueness theorem for implicit differential algebraic equations, Differential Integral Equations, 4 (1991), 563-582. Google Scholar

[12]

P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations, 109 (1994), 110-146. doi: 10.1006/jdeq.1994.1046. Google Scholar

[13]

P. J. Rabier and W. C. Rheinboldt, On impasse points of quasilinear differential algebraic equations, J. Math. Anal. Appl., 181 (1994), 429-454. doi: 10.1006/jmaa.1994.1033. Google Scholar

[14]

P. J. Rabier and W. C. Rheinboldt, On the computation of impasse points of quasilinear differential algebraic equations, Math. Comp., 62 (1994), 133-154. doi: 10.2307/2153400. Google Scholar

[15]

Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications World Scien. Publ. Co. Pte. Ltd. , 2008. doi: 10.1142/6746. Google Scholar

[16]

G. P. Veldes, J. Cuevas, P. G. Kevrekidis and D. J. Frantzeskakis, Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide Phys. Rev. E, 83 (2011), 046608. doi: 10.1103/PhysRevE. 83. 046608. Google Scholar

show all references

References:
[1]

F. Battelli and M. Fečkan, Nonlinear RLC circuits and implicit ODEs, Differential Integral Equations, 27 (2014), 671-690. Google Scholar

[2]

F. Battelli and M. Fečkan, Melnikov theory for nonlinear implicit ODEs, J. Differential Equations, 256 (2014), 1157-1190. doi: 10.1016/j.jde.2013.10.012. Google Scholar

[3]

F. Battelli and M. Fečkan, Melnikov theory for weakly coupled nonlinear RLC circuits Bound. Value Probl. , 2014: 101 (2014), 27pp. doi: 10.1186/1687-2770-2014-101. Google Scholar

[4]

F. Battelli and M. Fečkan, On the existence of solutions connecting singularities in nonlinear RLC circuits, Nonlinear Anal., 116 (2015), 26-36. doi: 10.1016/j.na.2014.12.015. Google Scholar

[5]

L. O. Chua, Ch. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits McGraw-Hill, New York, 1987.Google Scholar

[6]

E. Gluskin, A nonlinear resistor and nonlinear inductor using a nonlinear capacitor, J. Franklin Inst., 336 (1999), 1035-1047. doi: 10.1016/S0016-0032(99)00029-0. Google Scholar

[7]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution European Math. Soc. 2006. doi: 10.4171/017. Google Scholar

[8]

N. Lazarides, M. Eleftheriou and G. P. Tsironis, Discrete breathers in nonlinear magnetic metamaterials Phys. Rew. Lett. , 97 (2006), 157406. doi: 10.1103/PhysRevLett. 97. 157406. Google Scholar

[9]

M. Medved', Normal forms of implicit and observed implicit differential equations, Riv. Mat. Pura ed Appl., 10 (1991), 95-107. Google Scholar

[10]

M. Medved', Qualitative properties of generalized vector fields, Riv. Mat. Pura ed Appl., 15 (1994), 7-31. Google Scholar

[11]

P. J. Rabier and W. C. Rheinboldt, A general existence and uniqueness theorem for implicit differential algebraic equations, Differential Integral Equations, 4 (1991), 563-582. Google Scholar

[12]

P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations, 109 (1994), 110-146. doi: 10.1006/jdeq.1994.1046. Google Scholar

[13]

P. J. Rabier and W. C. Rheinboldt, On impasse points of quasilinear differential algebraic equations, J. Math. Anal. Appl., 181 (1994), 429-454. doi: 10.1006/jmaa.1994.1033. Google Scholar

[14]

P. J. Rabier and W. C. Rheinboldt, On the computation of impasse points of quasilinear differential algebraic equations, Math. Comp., 62 (1994), 133-154. doi: 10.2307/2153400. Google Scholar

[15]

Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications World Scien. Publ. Co. Pte. Ltd. , 2008. doi: 10.1142/6746. Google Scholar

[16]

G. P. Veldes, J. Cuevas, P. G. Kevrekidis and D. J. Frantzeskakis, Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide Phys. Rev. E, 83 (2011), 046608. doi: 10.1103/PhysRevE. 83. 046608. Google Scholar

Figure 1.  The heteroclinic orbits of equation (4.5)
Figure 2.  The heteroclinic orbits of equation (4.5)
[1]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87

[2]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784

[3]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353

[4]

Farid Tari. Two-parameter families of implicit differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139

[5]

Kunquan Lan. Eigenvalues of second order differential equations with singularities. Conference Publications, 2001, 2001 (Special) : 241-247. doi: 10.3934/proc.2001.2001.241

[6]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[7]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165

[8]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040

[9]

Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517

[10]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39

[11]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283

[12]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91

[13]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[14]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036

[15]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299

[16]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029

[17]

Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018

[18]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[19]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595

[20]

Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019012

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (10)
  • HTML views (26)
  • Cited by (1)

Other articles
by authors

[Back to Top]