zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A new analytical method for the linearization of dynamic equation on measure chains. (English) Zbl 1126.34030
The authors deal with dynamical systems on measure chains (time scales). Such systems (their definition is too complicated to be given here; see, for example, the book by {\it M. Bohner} and {\it A. Peterson} [Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001)] are generalizations of ordinary differential equations. In this paper, the authors introduce a new method for establishing topological equivalence on measure chains between a nonlinear system and its linear part.

MSC:
34C41Equivalence, asymptotic equivalence
39A12Discrete version of topics in analysis
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.; Bohner, M.: Basic calculus on time scales and some of its applications. Results math. 35, 3-22 (1999) · Zbl 0927.39003
[2] Agarwal, R. P.; Bohner, M.; O’regan, D.: Dynamic equations on time scales: A survey. J. comput. Appl. math. 141, No. 1 -- 2, 1-26 (2002) · Zbl 1020.39008
[3] Bohner, M.; Guseinov, G.; Peterson, A.: Introduction to the time scales calculus, advances in dynamic equations on time scales. (2003) · Zbl 1025.34001
[4] Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications. (2001) · Zbl 0978.39001
[5] Cao, J.; Li, Q.; Wan, S.: Periodic solutions of the higher dimensional non-autonomous system. Appl. math. Comput. 130, No. 2 -- 3, 369-383 (2002)
[6] Copple, W. A.: Dichotomies in stability theory. (1978)
[7] Erbe, L. H.; Peterson, A.: Green functions and comparison theorems for differential equations on measure chains. Dyn. contin. Discrete impuls. Syst. 6, 121-137 (1999) · Zbl 0938.34027
[8] Erbe, L.; Hilger, S.: Sturmain theory on measure chains. Differential equations dynam. Systems 1, 223-246 (1993) · Zbl 0868.39007
[9] Hale, J. K.: Ordinary differential equations. (1969) · Zbl 0186.40901
[10] Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[11] Hartman, P.: On the local linearization of differential equations. Proc. amer. Math. soc. 14, 568-573 (1963) · Zbl 0115.29801
[12] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Dissertation, Univ. Würzburg, 1988 · Zbl 0695.34001
[13] Hilger, S.: Analysis on measure chains a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001
[14] Hilger, S.: Generalized theorem of hartman -- grobman on measure chains. J. aust. Math. soc. Ser. A 60, No. 2, 157-191 (1996) · Zbl 0853.39007
[15] Hilger, S.: Smooth of invariant manifolds. J. funct. Anal. 106, 95-129 (1996) · Zbl 0762.93041
[16] Kaymakalan, B.; Lawrence, B. A.: Coupled solutions and monotone iterative techniques for some nonlinear initial value problems on time scales. Nonlinear anal. Real world appl. 2, 245-259 (2003)
[17] Kirchgraber, U.; Palmer, K. J.: Geometry in the neighborhood of an invariant manifolds of maps and flows and linearization. (1991) · Zbl 0746.58008
[18] Lakshmikantham, V.; Sivasundaram, S.; Kaymakçalan, B.: Dynamic systems on measure chains. Mathematics and its applications 370 (1996) · Zbl 0869.34039
[19] C. Pötzsche, Langsame Faserbündel Dynamischer Gleichungen auf Maßketten, PhD thesis, Logos Verlag, Berlin, 2002
[20] Pötzche, C.: Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients. J. math. Anal. appl. 289, 317-335 (2004)
[21] Pötzsche, C.: Exponential dichotomies for linear dynamic equations. Nonlinear anal. 47, 873-884 (2001) · Zbl 1042.34510
[22] Palmer, K. J.: A generalization of hartman’s linearization theorem. J. math. Anal. appl. 41, 753-758 (1973) · Zbl 0272.34056
[23] Shi, J.; Zhang, J.: Classification of the differential equations. (2003)
[24] Xia, Y. H.; Cao, J.: Almost periodicity in an ecological mode with M-predators and N-preys by ”pure-delay type” system. Nonlinear dynam. 39, No. 3, 275-304 (2005) · Zbl 1093.92061
[25] Y.H. Xia, J. Cao, Almost periodic solutions for an ecological model with infinite delays, Proc. Edinb. Math. Soc., in press · Zbl 1130.34044
[26] Xia, Y. H.; Lin, M.; Cao, J.: The existence of almost periodic solutions of certain perturbation system. J. math. Anal. appl. 310, No. 1, 81-96 (2005) · Zbl 1089.34039
[27] Yoshizawa, T.: Stability theory for the existence of periodic solutions and almost-periodic solutions. Applied mathematical sciences 14 (1975) · Zbl 0304.34051