Characterization for rectifiable and nonrectifiable attractivity of nonautonomous systems of linear differential equations. (English) Zbl 1294.34011

Summary: We study a new kind of asymptotic behaviour near \(t=0\) for the nonautonomous system of two linear differential equations: \[ x'(t)= A(t)x(t),\;t\in(0, t_0], \] where the matrix-valued function \(A= A(t)\) has a kind of singularity at \(t=0\). It is called rectifiable \(j\) (resp., nonrectifiable) attractivity of the zero solution, which means that \(\| x(t)\|_2\to 0\) as \(t\to 0\) and the length of the solution curve of \(x\) is finite (resp., infinite) for every \(x\neq 0\). It is characterized in terms of certain asymptotic behaviour of the eigenvalues of \(A(t)\) near \(t=0\).
Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at \(t=0\).


34A30 Linear ordinary differential equations and systems
37C60 Nonautonomous smooth dynamical systems
34D05 Asymptotic properties of solutions to ordinary differential equations
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[1] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons, 1999. · Zbl 1060.28005
[2] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, vol. 44 of Fractals and Rectifiability, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0880.28002
[3] C. Tricot, Curves and Fractal Dimension, Springer, New York, NY, USA, 1995. · Zbl 0866.11048
[4] S. Mili and M. Pa, “Nonautonomous differential equations in Banach space and nonrectifiable attractivity in two-dimensional linear differential systems,” Abstract and Applied Analysis, vol. 2013, Article ID 935089, 10 pages, 2013. · Zbl 1277.34086
[5] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass, USA, 1965. · Zbl 0154.09301
[6] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, New York, NY, USA, 3rd edition, 1971. · Zbl 0215.13802
[7] I. T. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, Izdatel’stvo Tbilisskogo Universiteta, Tbilisi, Ga, USA, 1975.
[8] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass, USA, 2nd edition, 1982. · Zbl 0476.34002
[9] M. Pinto, “Asymptotic integration of second-order linear differential equations,” Journal of Mathematical Analysis and Applications, vol. 111, no. 2, pp. 388-406, 1985. · Zbl 0591.34034
[10] J. Rovder, “Asymptotic and oscillatory behaviour of solutions of a linear differential equation,” Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 41-47, 1992. · Zbl 0764.34027
[11] S. Castillo and M. Pinto, “Asymptotic integration of ordinary different systems,” Journal of Mathematical Analysis and Applications, vol. 218, no. 1, pp. 1-12, 1998. · Zbl 0945.34035
[12] I. Rachunkova and I. Rachunek, “Asymptotic formula for oscillatory solutions of some singular nonlinear differential equation,” Abstract and Applied Analysis, vol. 2011, Article ID 981401, 9 pages, 2011. · Zbl 1222.34034
[13] J. Jaros and T. Kusano, “Existence and precise asymptotic behavior of strongly monotone solutions of systems of nonlinear differential equations,” Difference Equations and Their Applications, vol. 5, pp. 185-204, 2013. · Zbl 1294.34055
[14] M. Fe\vckan, Bifurcation and Chaos in Discontinuous and Continuous Systems, Nonlinear Physical Science, Springer, 2011.
[15] O. G. Mustafa and Y. V. Rogovchenko, “Asymptotic integration of a class of nonlinear differential equations,” Applied Mathematics Letters, vol. 19, no. 9, pp. 849-853, 2006. · Zbl 1126.34339
[16] M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, vol. 1907 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2007. · Zbl 1131.37001
[17] L. Hatvani, “On the asymptotic stability for a two-dimensional linear nonautonomous differential system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 9-10, pp. 991-1002, 1995. · Zbl 0844.34050
[18] J. Sugie and M. Onitsuka, “Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients,” Proceedings of the American Mathematical Society, vol. 138, no. 7, pp. 2493-2503, 2010. · Zbl 1207.34069
[19] F. Amato, M. Ariola, M. Carbone, and C. Cosentino, “Finite-time control of linear systems: a survey,” in Current Trends in Nonlinear Systems and Control, pp. 195-213, Birkhäuser, Boston, Mass, USA, 2006.
[20] G. Haller, “A variational theory of hyperbolic Lagrangian coherent structures,” Physica D, vol. 240, no. 7, pp. 574-598, 2011. · Zbl 1214.37056
[21] K. Rateitschak and O. Wolkenhauer, “Thresholds in transient dynamics of signal transduction pathways,” Journal of Theoretical Biology, vol. 264, no. 2, pp. 334-346, 2010.
[22] M. K. Kwong, M. Pa, and J. S. W. Wong, “Rectifiable oscillations in second-order linear differential equations,” Journal of Differential Equations, vol. 245, no. 8, pp. 2333-2351, 2008. · Zbl 1168.34027
[23] M. Pa, “Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 724-738, 2007. · Zbl 1126.34023
[24] M. Pa and J. S. W. Wong, “Rectifiable oscillations in second-order half-linear differential equations,” Annali di Matematica Pura ed Applicata. Series IV, vol. 188, no. 3, pp. 517-541, 2009. · Zbl 1184.34043
[25] M. Pa and S. Tanaka, “Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 27-42, 2011. · Zbl 1223.34047
[26] J. S. W. Wong, “On rectifiable oscillation of Euler type second order linear differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 20, pp. 1-12, 2007. · Zbl 1182.34049
[27] J. S. W. Wong, “On rectifiable oscillation of Emden-Fowler equations,” Memoirs on Differential Equations and Mathematical Physics, vol. 42, pp. 127-144, 2007. · Zbl 1157.34027
[28] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York, NY, USA, 1999. · Zbl 0954.49024
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