On the Perron Effect for Exponential Stability of Differential Systems on Time Scales
Main Article Content
Abstract
In 2007, N. H. Du and L. H. Tien [1] shown that the exponential stability of the linear equation on time scales implies the exponential stability of the suitable small enough Lipchitz perturbed equation. In this paper, we shall prove that if the perturbation is arbitrary small order 1 then the above argument is not true which is called Perron effect.
Keywords:
Exponential stability, Perron effect, time scales, linear dynamic equation.
References
[1] N. H. Du and L. H. Tien, On the exponential stability of dynamic equations on time scales, J. Math. Anal. Appl, 331(2007), 1159 - 1174.
[2] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18--56.
[3] W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics No. 29, Springer-Verlag, Berlin, 1978.
[4] A. Peterson, R. F. Raffoul, Exponential stability of dynamic equations on time scales, Adv. Difference Equ. Appl. (2005) 133--144.
[5] C. Potzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003) 1223-1241.
[6] C. Potzsche, Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, J. Anal. Math. Appl. 289 (2004) 317-335.
[7] O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Mathematische Zeitschrift, 1930, vol. 32, no. 5, pp. 702 - 728.
[8] S. K. Korovin, N. A. Izobov, On the Perron Sign Change Effect for Lyapunov Characteristic Exponents of Solutions of Differential Systems, 2010, Diff. Equa, vol. 46, no. 10, pp. 1395 - 1408.
[9] N. V. Kuznetsov, G. A. Leonov, Counterexample of Perron in the Discrete Case, Izv. RAEN, Diff. Uravn., No 5, 2001, pp. 71.
[10] M. Bohner and A. Peterson, Dynamic equations on time scales: an introduction with applications, Birkhauser Boston, Inc., 2001.
[11] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhauser Boston, Inc., Boston, 2003.
[12] J. Zhang, M. Fanb, H. Zhu, Necessary and sufficient criteria for the existence of exponential dichotomy on time scales, Computers and Mathematics with Applications 60 (2010) 2387–2398.
[13] A. Cabada and D. R. Vivero, Expression of the Lebesgue -integral on time scales as a usual Lebesgue integral; application to the calculus of -antiderivatives, Mathematical and Computer Modelling 43 (2006) 194–207.
[2] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18--56.
[3] W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics No. 29, Springer-Verlag, Berlin, 1978.
[4] A. Peterson, R. F. Raffoul, Exponential stability of dynamic equations on time scales, Adv. Difference Equ. Appl. (2005) 133--144.
[5] C. Potzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003) 1223-1241.
[6] C. Potzsche, Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, J. Anal. Math. Appl. 289 (2004) 317-335.
[7] O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Mathematische Zeitschrift, 1930, vol. 32, no. 5, pp. 702 - 728.
[8] S. K. Korovin, N. A. Izobov, On the Perron Sign Change Effect for Lyapunov Characteristic Exponents of Solutions of Differential Systems, 2010, Diff. Equa, vol. 46, no. 10, pp. 1395 - 1408.
[9] N. V. Kuznetsov, G. A. Leonov, Counterexample of Perron in the Discrete Case, Izv. RAEN, Diff. Uravn., No 5, 2001, pp. 71.
[10] M. Bohner and A. Peterson, Dynamic equations on time scales: an introduction with applications, Birkhauser Boston, Inc., 2001.
[11] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhauser Boston, Inc., Boston, 2003.
[12] J. Zhang, M. Fanb, H. Zhu, Necessary and sufficient criteria for the existence of exponential dichotomy on time scales, Computers and Mathematics with Applications 60 (2010) 2387–2398.
[13] A. Cabada and D. R. Vivero, Expression of the Lebesgue -integral on time scales as a usual Lebesgue integral; application to the calculus of -antiderivatives, Mathematical and Computer Modelling 43 (2006) 194–207.