August  2016, 9(4): 895-922. doi: 10.3934/dcdss.2016034

Blue sky-like catastrophe for reversible nonlinear implicit ODEs

1. 

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

2. 

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

Received  March 2015 Revised  June 2015 Published  August 2016

We study for reversible implicit differential equations the bifurcation of bounded solutions connecting singularities in finite time and their approximation by shadowed periodic solutions. Melnikov like condition is derived. Application is given to planar nonlinear RLC system.
Citation: Flaviano Battelli, Michal Fečkan. Blue sky-like catastrophe for reversible nonlinear implicit ODEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 895-922. doi: 10.3934/dcdss.2016034
References:
[1]

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

[2]

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

[3]

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

[4]

A. W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., Vol. 629, Springer-Verlag, Berlin, 1978.  Google Scholar

[5]

R. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J., 26 (1977), 247-263. doi: 10.1512/iumj.1977.26.26018.  Google Scholar

[6]

M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford Ser. (2), 23 (1971), 113-133. doi: 10.1093/qmath/23.2.113.  Google Scholar

[7]

E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[8]

X. B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Royal Soc. Edinburgh A, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.  Google Scholar

[9]

K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differential Equations, 65 (1986), 321-360. doi: 10.1016/0022-0396(86)90023-9.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

________, On the computation of impasse points of quasilinear differential algebraic equations, Math. Comp., 62 (1994), 133-154. doi: 10.2307/2153400.  Google Scholar

[14]

R. Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[15]

A. Vanderbauwhede, Heteroclinic cycles and periodic orbits in reversible systems, in Ordinary and Delay Differential Equations, (eds. J. Wiener and J.K. Hale), Pitman Res. Notes Math. Ser., 272, Longman Sci. Tech., Harlow, (1992), 250-253.  Google Scholar

[16]

A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292-318. doi: 10.1007/BF00946632.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

A. W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., Vol. 629, Springer-Verlag, Berlin, 1978.  Google Scholar

[5]

R. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J., 26 (1977), 247-263. doi: 10.1512/iumj.1977.26.26018.  Google Scholar

[6]

M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford Ser. (2), 23 (1971), 113-133. doi: 10.1093/qmath/23.2.113.  Google Scholar

[7]

E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[8]

X. B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Royal Soc. Edinburgh A, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.  Google Scholar

[9]

K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differential Equations, 65 (1986), 321-360. doi: 10.1016/0022-0396(86)90023-9.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

________, On the computation of impasse points of quasilinear differential algebraic equations, Math. Comp., 62 (1994), 133-154. doi: 10.2307/2153400.  Google Scholar

[14]

R. Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[15]

A. Vanderbauwhede, Heteroclinic cycles and periodic orbits in reversible systems, in Ordinary and Delay Differential Equations, (eds. J. Wiener and J.K. Hale), Pitman Res. Notes Math. Ser., 272, Longman Sci. Tech., Harlow, (1992), 250-253.  Google Scholar

[16]

A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292-318. doi: 10.1007/BF00946632.  Google Scholar

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