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

Flaviano Battelli; Michal Fečkan

doi:10.3934/dcdsb.2017162

Article Contents

2017, Volume 22, Issue 8: 3043-3061. Doi: 10.3934/dcdsb.2017162

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

Flaviano Battelli^1,1, and
Michal Fečkan^2,3,2,

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: October 2017

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Abstract

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.

Keywords:

Implicit ordinary differential equations,
impasse points,
IK-singularities,
RLC circuits.

Mathematics Subject Classification: 34A09, 37C60, 47N70.

Citation:

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Figure 1. The heteroclinic orbits of equation (4.5)

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Figure 2. The heteroclinic orbits of equation (4.5)

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References

[1]	F. Battelli and M. Fečkan, Nonlinear RLC circuits and implicit ODEs, Differential Integral Equations, 27 (2014), 671-690.
[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.
[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.
[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.
[5]	L. O. Chua, Ch. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits McGraw-Hill, New York, 1987.
[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.
[7]	P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution European Math. Soc. 2006. doi: 10.4171/017.
[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.
[9]	M. Medved', Normal forms of implicit and observed implicit differential equations, Riv. Mat. Pura ed Appl., 10 (1991), 95-107.
[10]	M. Medved', Qualitative properties of generalized vector fields, Riv. Mat. Pura ed Appl., 15 (1994), 7-31.
[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.
[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.
[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.
[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.
[15]	Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications World Scien. Publ. Co. Pte. Ltd. , 2008. doi: 10.1142/6746.
[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.