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Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems. (English) Zbl 1256.34028
Imagine a one-parameter family of piecewise smooth planar systems of ordinary differential equations for which a “corner” that is formed by the intersection of finitely many curves that bound the domains of smoothness is a singularity for all values of the parameter. One can then consider the possibility of the creation of a limit cycle from the singularity as the parameter is varied, i.e., a “generalized Hopf bifurcation”. The authors examine this situation and prove that even when the eigenvalues of the linear parts of all the individual smooth systems that meet at the corner are real (and nonzero and distinct) a generalized Hopf bifurcation can occur. They provide a numerical example illustrating the phenomenon. They also examine the situation when the eigenvalues of some of the linear parts involved are as above, but others are complex conjugates. In each case treated, the corner has a finite number $n$ of boundary rays emanating from it, determining $n$ sectors, on each of which the system is at least $C^3$ and has a linear part that depends analytically on the parameter $\lambda \in {\mathbb R}$.

MSC:
 34C23 Bifurcation (ODE) 34A36 Discontinuous equations
Full Text:
References:
 [1] Brogliato, B.: Impacts in mechanical systems- analysis and modelling, Lecture notes in physics 551 (2000) · Zbl 0972.00062 [2] Di Bernardo, M.; Budd, C. J.; Champneys, A. R.; Kowalczyk, P.: Piecewise-smooth dynamical systems: theory and application, Appl. math. Sci. series 163 (2008) · Zbl 1146.37003 [3] Kunze, M.; Marques, M. M.: Impacts in mechanical systems, (2000) [4] Leine, R. I.; Nijmeijer, H.: Dynamics and bifurcations in non-smooth mechanical systems, (2004) · Zbl 1068.70003 [5] Chin, W.; Ott, E.; Nusse, H.; Grebogi, C.: Grazing bifurcations in imp- act oscillators, Phys. rev. E 50, 4427-4444 (1994) [6] Nordmark, A.: Non-periodic motion caused by grazing incidence in an impact oscillator, J. sound vib. 145, 279-297 (1991) [7] Budd, C. J.; Dux, F.: Chattering and related behaviour in impact oscillators, Philos. trans. R. soc. Lond. A 347, 365-389 (1994) · Zbl 0816.70018 · doi:10.1098/rsta.1994.0049 [8] Nordmark, A. B.; Piiroinen, P. T.: Simulation and stability analysis of impacting systems with complete chattering, Nonlinear dyn. 58, 85-106 (2009) · Zbl 1183.70038 · doi:10.1007/s11071-008-9463-y [9] Di Bernardo, M.; Johansson, K. H.; Vasca, F.: Self-oscillations and sliding in relay feedback systems: symmetry and bifurcations, Int. J. Bifurcation chaos 11, No. 4, 1121-1140 (2001) [10] Filippov, A. F.: Differential equations with discontinuous righthand sides, (1988) · Zbl 0664.34001 [11] Budd, C. J.; Piiroinen, P. T.: Corner bifurcations in non-smoothly forced impact oscillators, Phys. D 220, 127-145 (2006) · Zbl 1113.37060 · doi:10.1016/j.physd.2006.07.001 [12] Di Bernardo, M.; Budd, C. J.; Champneys, A. R.: Corner collision implies border-collision bifurcation, Phys. D 154, 171-194 (2001) · Zbl 0984.34028 · doi:10.1016/S0167-2789(01)00250-0 [13] Di Bernardo, M.; Feigin, M. I.; Hogan, S. J.; Homer, M. E.: Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems, Chaos, solitons and fractals 10, 1881-1908 (1999) · Zbl 0967.37030 · doi:10.1016/S0960-0779(98)00317-8 [14] Feigin, M. I.: Doubling of the oscillation period with C-bifurcations in piecewise-continuous systems, Prikl. mat. Mekh. 34, 861-869 (1970) · Zbl 0224.34021 · doi:10.1016/0021-8928(70)90064-X [15] Diamond, P.; Rachinskii, D.; Yumagulov, M.: Stability of large cycles in a nonsmooth problemwith Hopf bifurcation at infinity, Nonlinear anal. 42, 1017-1031 (2000) · Zbl 0963.34034 · doi:10.1016/S0362-546X(99)00162-5 [16] Diamond, P.; Kuznetsov, N.; Rachinskii, D.: On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity, J. differential equations 175, No. 1, 1-26 (2001) · Zbl 0984.34029 · doi:10.1006/jdeq.2000.3916 [17] Huan, S. M.; Yang, X. -S.: On the number of limit cycles in general planar piecewise linear systems, Discrete contin. Dyn. ser.-A 32, No. 6, 2147-2164 (2012) · Zbl 1248.34033 [18] Küpper, T.; Mortitz, S.: Generalized Hopf bifurcation for non-smooth planar systems, Philos. trans. R. soc. Lond. ser. A 359, 2483-2496 (2001) · Zbl 1097.37502 · doi:10.1098/rsta.2001.0905 [19] Leine, R. I.; Van Campen, D. H.: Bifurcation phenomena in non-smooth dynamical systems, Eur. J. Mech. A: solids 25, 595-616 (2006) · Zbl 1187.70041 · doi:10.1016/j.euromechsol.2006.04.004 [20] Simpson, D. J. W.; Meiss, J. D.: Andronov--Hopf bifurcations in planar piecewise-smooth continuous flows, Phys. lett. A 371, 213-220 (2007) · Zbl 1209.37059 · doi:10.1016/j.physleta.2007.06.046 [21] Zou, Y.; Küpper, T.; Beyn, W. -J.: Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. nonlinear sci. 16, 159-177 (2006) · Zbl 1104.37031 · doi:10.1007/s00332-005-0606-8 [22] Akhmet, M. U.: Perturbations and Hopf bifurcation of the planar disconti- nuous dynamical system, Nonlinear anal. 60, 163-178 (2005) · Zbl 1066.34008 · doi:10.1016/j.na.2004.08.029 [23] Akhmet, M. U.; Arugaslan, D.: Bifurcation of a non-smooth planar limit cycle from a vertex, Nonlinear anal. 71, 2723-2733 (2009) [24] Huan, S. M.; Yang, X. -S.: Generalized Hopf bifurcation in a class of planar switched system, Dyn. syst. 26, No. 4, 433-445 (2011) · Zbl 1252.37041 [25] Zou, Y.; Küpper, T.: Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems, Nonlinear anal. TMA 62, No. 1, 1-17 (2005) · Zbl 1084.34006 · doi:10.1016/j.na.2004.06.004 [26] Zhang, J. Y.; Feng, B. Y.: Geometric theory of ordinary differential equations and bifurcation problems, (2000)