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On the limit cycles of a class of piecewise linear differential systems in 4 with two zones. (English) Zbl 1251.37027

The authors study the maximum number of limit cycles of the 4-dimensional continuous piecewise linear vector field

x ' =A 0 x+εF(x),

where ε is a small parameter, A 0 is an elliptic matrix with eigenvalues +-i (in the Jordan normal form) and F: 4 4 is given by F(x)=Ax+φ(k T x)b for a matrix A, k,b 4 {0} and φ: is the piecewise linear function such that φ(s)=0 for s(,1) and φ(s)=ms for s[1,).

The main result is that the upper bound for the number of limit cycles of the above defined system is three, and there are systems having exactly three limit cycles. The proof is based on the non-smooth averaging theory.

37C10Vector fields, flows, ordinary differential equations
34C23Bifurcation (ODE)
34A36Discontinuous equations
[1]A., Buică; J., Llibre: Bifurcation of limit cycles from a four-dimensional center in control systems, Int. J. Bifurcat. chaos appl. Sci. eng. 15, 2653-2662 (2005) · Zbl 1092.34519 · doi:10.1142/S0218127405013599
[2]A., Buică; J., Llibre: Averaging methods for finding periodic orbits via brower degree, Bull. sci. Math. 128, 7-22 (2004) · Zbl 1055.34086 · doi:10.1016/j.bulsci.2003.09.002
[3]C., Christopher; C., Li: Limit cycles of differential equations, advanced courses in mathematics – CRM Barcelona, (2007)
[4]E., Freire; E., Ponce; F., Rodrigo; F., Torres: Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcat. chaos appl. Sci. eng. 8, 2073-2097 (1998) · Zbl 0996.37065 · doi:10.1142/S0218127498001728
[5]J., Guckenheimer; P., Holmes: Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, (1983)
[6]S., Ilyashenko Y.: Centennial history of Hilbert’s 16th problem, Bull. am. Math. soc. 39, 301-354 (2002) · Zbl 1004.34017 · doi:10.1090/S0273-0979-02-00946-1
[7]J., Li: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcat. chaos appl. Sci. eng. 13, 47-106 (2003) · Zbl 1063.34026 · doi:10.1142/S0218127403006352
[8]G., Lloyd N.: Degree theory, (1978)
[9]R., Lum; O., Chua L.: Global properties of continuous piecewise-linear vector fields. Part I: Simplest case in R2, memorandum UCB/ERL M90/22, (1990)
[10]A., Sanders J.; F., Verhulst: Averaging methods in nonlinear dynamical systems, applied mathematical sciences, vol. 59, (1985) · Zbl 0586.34040
[11]F., Verhulst: Nonlinear differential equations and dynamical systems, universitext, (1991)