Differential systems under small perturbations
Keywords:
linear differential systems, singular exponents, nonlinear differential systems, bound of the solutionsAbstract
The stability, changes and mobility bounds of generalized Lyapunov exponents of linear homogeneous systems of differential equations with continuous and tending to zero coefficients under small perturbations was studied with respect to the generalized central and generalized singular exponents. An example of the instability of generalized Lyapunov exponents, under small perturbations tends to zero was given. Using the generalized upper central exponents the exact upper limit of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using the generalized lower central exponents the exact lower bounds of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using a special upper generalized exponent the upper bounds of change of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Using the generalized lower special exponent the lower bounds of changes of generalized Lyapunov exponents of linear systems under small perturbations was determined in a certain class of nonlinear systems of differential equations. Mutual relations between generalized upper and lower central exponents, upper and lower generalized singular exponents was established. A brief description of the generalized Lyapunov exponents of the generalized upper and lower central exponents, generalized upper and lower singular exponents as Baire functions in a certain metric space was given.
References
[2] Atkinson F.V. "On stability and asymptotic equilibrium". Ann. Math., vol. 68 (1958) : 690-708.
[3] Bellman R. "On the boundedness of solutions of nonlinear differential and difference equations (6, 8)’". Trans. Amer. Math. Soc., vol 62 (1947) : 357-386.
[4] Bihari I. "A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations (3, 6)". Acta Math. Acad Sci. Hung., vol 7 (1956) : 71-94.
[5] Coddington E.A. and Levinson N. Theory of differential equations (McGraw-Hill, 1955), 149
[6] Conti R. "Systemi differenziali asintoticamente equivalent’". Rend. Lincel., Ser. VIII, vol. 22, No 5 (1957) : 588-592.
[7] Diliberto S.P. and Hufford G. "Perturbation theorems for non-linear ordinary differential equations". Ann. Math. Studies, Vol. 8 (1956) : 207-236.
[8] Hartman P., Wintner A. "Asymptotic integrations of ordinary nonlinear differential equations". IBID, vol. 77 [VIII.3; X.4, 8, 11, 13] (1955) : 692-724
[9] Levinson N. "On stability of nonlinear systems of differential equations". Colloq. Math., vol. 6 (1949) : 40-45.
[10] Massera J.L. "On Lyapunov’s conditions of stability (6, 7)". Ann. Math., vol. 2 (1949) : 705-721.
[11] O. Perron. "Die Stabilitatsfrage bei Differentialgleichungen". Math. Z., vol. 32 (1930) : 703-728.
[12] Aldibekov T.M., Mirzakulova A.E., Aldazharova M.M. "Ob ustoichivyh asimptoticheskih harakteristikah differentsialnyh sistem. [Stable asymptotical characteristics of differential systems]"Bulletin of KazNU. Math., Mech., Inf. Series, vol. 2 (2015) : 33-41.
[13] Aldibekov T.M. "Obobshiennye osobye pokazateli lineinyh sistem differentsialnyh uravnenii. [Generalized special performance linear system of differential equations]". Bulletin of KazNU. Math., Mech., Inf. Series, vol. 1 (2016) : 47-54.
[14] Aldibekov T.M. "Ob ocenkah reshenii differentsialnyh sistem. [Estimates of solutions of differential systems]". International scientific conference "Modern problems of Mathematics, Mechanics and Informatics"dedicated to the 25th anniversary of Independence of the Republic of Kazakhstan. Karaganda, December 9-10, 2016.
[15] Aldibekov T.M., Moldabek Z.T, Mirzakulova A.E., Aldazharova M.M. "O ravnomenoi otsenke snizu dlya reshenii nelineinyh sistem differentsialnyh uravnenii [The uniform lower bound for solutions of nonlinear system of differential equations]". Bulletin of KazNU. Math., Mech., Inf. Series, vol. 4 (2016) : 46-54
[16] Vinograd R.E. "Otsenka skachka pri malyh vozmusheniyah [Bounds of the jump function on small perturbations]". Doklady AN SSR, vol. 114 (1957) : 459-461.
[17] Millionschikov V.M. "Grubye svoistva differentcialnyh uravnenii[Rough haracters of differential equations]". Diff. uravn, vol. 5 (1969) : 1775-1784.
[18] Millionschikov V.M. "Dokazatelstva dostizhimosti tsentralnyh pokazatelei lineinyh sistem diff. ur. [Proof of the approachability of cental exponents]". Uspehi mat. nauk, vol. 23 (1968).
[19] Millionschikov V.M. "Dokazatelstva dostizhimosti tsentralnyh pokazatelei lineinyh sistem diff. ur. [Proof of the approachability of cental exponents]".Sib.mat.jur., vol. 10 (1969) : 99-104.
[20] Millionschikov V.M. "Berovskie klassy funktsii i pokazateli Lyapunova. XII [Baire classes of functions and Lyapunov exponents. XII]". Diff.uravn., vol. 19 (1983) : 196-220.
[21] Millionschikov V.M. "O tipichnyh svoistvah uslovnoi eksponentsialnoi ustoichivosti [Typical parameters of conditional stability]". Diff.uravn., vol. 19 (1983) : 1344-1356.
