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)
Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems. (English) Zbl 1131.34030
Summary: We consider a three-parameter class of Liénard type nonlinear dissipative systems of the form x ¨+(b+3kx)x ˙+k 2 x 3 +bkx 2 +λx=0. Since such dissipative systems admit an eight-parameter Lie group of point transformations, it follows that there exists a (complex) point transformation mapping such a system into the free particle system x ¨=0. Normally, such an explicit point transformation cannot be found. Here we find such an explicit point transformation through exploiting the group properties of the determining equations that lead to it. Consequently, we obtain the explicit general solution of such dissipative systems. Moreover, we completely characterize the asymptotic and/or finite time blow-up behaviour of such systems in terms of their three parameters and initial data.
MSC:
34C14Symmetries, invariants (ODE)
34A05Methods of solution of ODE
References:
[1]Chandrasekhar, S.: An introduction to the study of stellar structure, (1957) · Zbl 0079.23901
[2]Dixon, J. M.; Tuszynski, J. A.: Solutions of a generalized Emden equation and their physical significance, Phys. rev. A 41, 4166-4173 (1990)
[3]Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.: Unusual Liénard-type nonlinear oscillators, Phys. rev. E 72, 066203 (2005)
[4]V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, A nonlinear oscillator with unusual dynamical properties, in: Proc. 3rd National Conference on Nonlinear Systems and Dynamics, Allied Publishers, Chennai, 2006, pp. 1 – 4
[5]Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.: On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. lond. Ser. A 461, 2451-2476 (2005) · Zbl 1186.34046 · doi:10.1098/rspa.2005.1465
[6]Chandrasekar, V. K.; Pandey, S. N.; Senthilvelan, M.; Lakshmanan, M.: A simple and unified approach to identify integrable nonlinear oscillators and systems, J. math. Phys. 47, 023508 (2006) · Zbl 1111.34003 · doi:10.1063/1.2171520
[7]Chandrasekar, V. K.; Senthilvelan, M.; Kundu, A.; Lakshmanan, M.: A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators, J. phys. A 39, 9743-9754 (2006) · Zbl 1107.34003 · doi:10.1088/0305-4470/39/31/006
[8]Mahomed, F. M.; Leach, P. G. L.: The linear symmetries of a nonlinear differential equation, Quaest. math. 8, 241-274 (1985) · Zbl 0618.34009 · doi:10.1080/16073606.1985.9631915
[9]Ince, E. L.: Ordinary differential equations, (1956)
[10]Prelle, M. J.; Singer, M. F.: Elementary first integrals of differential equations, Trans. amer. Math. soc. 279, 215-229 (1983) · Zbl 0527.12016 · doi:10.2307/1999380
[11]Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989)
[12]Bluman, G. W.: Invariant solutions for ordinary differential equations, SIAM J. Appl. math. 50, 1706-1715 (1990) · Zbl 0714.34008 · doi:10.1137/0150101
[13]Bluman, G. W.; Anco, S. C.: Symmetry and integration methods for differential equations, (2002)
[14]Minorsky, N.: Nonlinear oscillations, (1962) · Zbl 0102.30402
[15]Koga, T.; Shinagawa, M.: An extension of the Liénard theorem and its application, IEEE internat. Sympos. circuits syst. 2, 1244-1247 (1991)
[16]Bluman, G. W.: On the transformation of diffusion processes into the Wiener process, SIAM J. Appl. math. 39, 238-247 (1980) · Zbl 0448.60056 · doi:10.1137/0139021
[17]Mahomed, F. M.; Leach, P. G. L.: The Lie algebra sl(3,R) and linearization, Quaest. math. 12, 121-139 (1989) · Zbl 0683.34004 · doi:10.1080/16073606.1989.9632170