zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems. (English) Zbl 1232.34047

The paper is concerned with the dynamics of a system

𝐱 ˙=A𝐱-𝐛inS - Γ - ,𝐱 ˙=B𝐱inΓ - S 0 Γ + ,𝐱 ˙=A𝐱+𝐛inS + Γ + ,

where S 0 =x,y,z 3 :-1<x<1, S ± =x,y,z 3 :±x>1, and

Γ ± =x,y,z 3 :x=±1, B=A+𝐛𝐜 T , A is a constant 3×3 matrix, 𝐛,𝐜 n · This is a particular case of the Lur’e system in control theory. The authors develop algebraic tools for the study of symmetric periodic orbits using closing equations for symmetric periodic orbits. The principal result in the paper determines all the symmetric periodic orbits having two points in the plane Γ + . Computations leading to the first harmonic bifurcation diagram are provided. In the final part of the paper, a particular case of the system under study, a dimensionless Chua oscillator is studied.

34C05Location of integral curves, singular points, limit cycles (ODE)
34C25Periodic solutions of ODE
[1]Atherton, D. P.: Nonlinear control engineering, Describing function analysis and design (1982) · Zbl 0485.93001
[2]Barnett, S.; Cameron, R. G.: Introduction to mathematical control theory, (1985)
[3]Khalil, H. K.: Nonlinear systems, (1992) · Zbl 0969.34001
[4]Mees, A. I.: Dynamics of feedback systems, (1981) · Zbl 0454.93003
[5]Vidyasagar, M.: Nonlinear systems: analysis, (1993) · Zbl 0900.93132
[6]Lozano, R.; Brogliato, B.; Egeland, D.; Maschke, B.: Dissipative systems, analysis and control, theory and applications, (2000)
[7]Barabanov, N. E.: On the Kalman problem, Sib. math. J. 29, 333-341 (1988) · Zbl 0713.93044 · doi:10.1007/BF00969640
[8]Bernat, J.; Llibre, J.: Counterexample to Kalman and markus–yamabe conjectures in dimension larger than 3, Dyn. contin. Discrete impuls. Syst. 2, 337-379 (1996) · Zbl 0889.34047
[9]Moreno, I.; Suárez, R.: Existence of periodic orbits of stable saturated systems, Systems control lett. 51, 293-309 (2004) · Zbl 1157.93409 · doi:10.1016/j.sysconle.2003.09.004
[10]Ponce, E.; Ros, J.: On periodic orbits of 3D symmetric piecewise linear systems with real triple eigenvalues, Internat. J. Bifur. chaos 19, 2391-2399 (2009) · Zbl 1176.34048 · doi:10.1142/S0218127409024165
[11]Kennedy, M. P.: Three steps to chaos–part II: a Chua’s circuit premier, IEEE trans. Circuits syst. I fundam. Theory appl. 40, 657-674 (1993) · Zbl 0844.58059 · doi:10.1109/81.246141
[12]Andronov, A. A.; Vitt, A. A.; Khaikin, S. E.: Theory of oscillators, (1966) · Zbl 0188.56304
[13]Liu, L.; Wong, Y. S.; Lee, B. H. K.: Nonlinear aeroelastic analysis using the point transformation method, part I: Freeplay model, J. sound vib. 253, 447-469 (2002)
[14]Allwright, D. J.: Harmonic balance and the Hopf bifurcation theorem, Math. proc. Cambridge philos. Soc. 82, 453-467 (1977) · Zbl 0365.34040 · doi:10.1017/S0305004100054128
[15]Aguirre, B.; Alvarez-Ramirez, J.; Fernandez, G.; Suarez, R.: First harmonic analysis of linear control systems with high-gain saturating feedback, Internat. J. Bifur. chaos 7, 2501-2510 (1997) · Zbl 0967.93505 · doi:10.1142/S0218127497001679
[16]Alvarez, J.; Curiel, L. E.: First harmonic analysis of linear control systems with high-gain saturating feedback, Internat. J. Bifur. chaos 7, No. 8, 1811-1822 (1997)
[17]Basso, M.; Genesio, R.: Analysis and control of limit cycle bifurcations, Lncis 293, 127-154 (2003) · Zbl 1032.93523
[18]Llibre, J.; Ponce, E.: Global first harmonic bifurcation diagram for odd piecewise linear control systems, Dyn. stab. Syst. 11, 49-88 (1996) · Zbl 0855.93037 · doi:10.1080/02681119608806216
[19]Rapp, P. E.; Mees, A. I.: Spurius prediction of limit cycles in a non-linear feedback system by the describing function method, Internat. J. Control 26, 821-829 (1977) · Zbl 0372.93027 · doi:10.1080/00207177708928526
[20]Freire, E.; Ponce, E.; Ros, J.: The focus-center-limit cycle bifurcation in symmetric 3D piecewise linear systems, SIAM J. Appl. math. 65, 1933-1951 (2005) · Zbl 1080.37057 · doi:10.1137/040606107
[21]Lang, S.: Algebra, (1993) · Zbl 0848.13001
[22]Olver, P.: Classical invariant theory, London math. Soc. student texts 44 (1999)
[23]Carmona, V.; Freire, E.; Ponce, E.; Ros, J.; Torres, F.: Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones. Application to Chua’s circuit, Internat. J. Bifur. chaos 15, 2469-2484 (2005) · Zbl 1092.37520 · doi:10.1142/S0218127405013423