## A theory for non-smooth dynamic systems on the connectable domains.(English)Zbl 1065.34007

This paper studies a certain class of nonsmooth dynamical systems in the plane. Let $$\Omega$$ be a region of $$\mathbb{R}^2$$, decomposed as the disjoint union $$\Omega = \bigcup_{i=1}^n \Omega_i$$. Consider a dynamical system $$\dot{x} = F (x,t;\mu)$$ on $$\Omega$$, depending on the parameters $$\mu \in \mathbb{R}^n$$. It is assumed that $$F$$ is smooth and Lipschitz in each of the $$\Omega_i$$ (call $$F^{(i)}$$ the restriction of $$F$$ to $$\Omega_i$$), and $$\Xi \subset \Omega$$ is the set on which $$F$$ cannot be expressed as a piecewise smooth and continuous vector function. The subdomains $$\Omega_i$$ are termed accessible, while $$\Xi$$ is termed inaccessible; $$\Omega$$ is said to be connectable if the union of the $$\Omega_i$$ is connected, and separable otherways (these terms are not standard).
With this notation, the paper studies nonsmooth dynamical systems on connectable subdomains. First of all, properties of separating boundaries based on the characteristics of the flows $$F^{(i)}$$ are determined. Then, the local singularity and transversality properties – and correspondingly, the bouncing and tangency of solutions – of a flow at the boundaries between subdomains $$\Omega_i$$ are studied. In particular, sufficient and necessary conditions for a local singularity, transversality and bouncing of flows are investigated. The sliding dynamics at boundaries is also considered.

### MSC:

 34A36 Discontinuous ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations

### Keywords:

nonsmooth dynamical systems; transversality; bifurcations
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### References:

 [1] Masri, S.F.; Caughey, T.D., On the stability of the impact damper, ASME J. appl. mech., 33, 586-592, (1966) [2] Masri, S.F., General motion of impact dampers, J. acoust. soc. am., 47, 229-237, (1970) [3] Den Hartog, J.P., Forced vibration with combined viscous and Coulomb damping, Phil. mag., VII, 9, 801-817, (1930) · JFM 56.1215.03 [4] Den Hartog, J.P.; Mikina, S.J., Forced vibrations with non-linear spring constants, ASME J. appl. mech., 58, 157-164, (1932) [5] Utkin, V.I., Sliding modes and their application in variable structure systems, (1978), Mir Moscow · Zbl 0398.93003 [6] Utkin, V.I., Sliding regimes in optimization and control problem, (1981), Nauka Moscow [7] Filippov, A.F., Differential equations with discontinuous righthand sides, (1988), Kluwer Academic Publishers Dordrecht [8] Ye, H.; Michel, A.; Hou, L., Stability theory for hybrid systems, IEEE trans. automat. contr., 43, 4, 461-474, (1998) · Zbl 0905.93024 [9] Broucke, M.; Pugh, C.; Simic, S.N., Structural stability of piecewise smooth systems, Comput. appl. math., 20, 1-2, 51-89, (2001) · Zbl 1121.37307 [10] Senator, M., Existence and stability of periodic motions of a harmonically forced impacting system, J. acoust. soc. am., 47, 1390-1397, (1970) [11] Bapat, C.N.; Popplewell, N.; Mclachlan, K., Stable periodic motion of an impact pair, J. sound vib., 87, 19-40, (1983) · Zbl 0556.70019 [12] Shaw, S.W.; Holmes, P.J., A periodically forced impact oscillator with large dissipation, ASME J. appl. mech., 50, 849-857, (1983) · Zbl 0539.70032 [13] Luo ACJ. Analytical Modeling of Bifurcations, Chaos and Fractals in Nonlinear Dynamics. PhD Dissertation, University of Manitoba, Winnipeg, Canada, 1995 [14] Han, R.P.S.; Luo, A.C.J.; Deng, W., Chaotic motion of a horizontal impact pair, J. sound vib., 181, 231-250, (1995) · Zbl 1237.70028 [15] Luo, A.C.J., An unsymmetrical motion in a horizontal impact oscillator, ASME J. vib. acoust., 124, 420-426, (2002) [16] Shaw, S.W.; Holmes, P.J., A periodically forced piecewise linear oscillator, J. sound vib., 90, 1, 121-155, (1983) · Zbl 0561.70022 [17] Natsiavas, S., Periodic response and stability of oscillators with symmetric trilinear restoring force, J. sound vib., 134, 2, 315-331, (1989) · Zbl 1235.70094 [18] Nordmark, A.B., Non-periodic motion caused by grazing incidence in an impact oscillator, J. sound vib., 145, 279-297, (1991) [19] Kleczka, M.; Kreuzer, E.; Schiehlen, W., Local and global stability of a piecewise linear oscillator, Phil. trans.: phys. sci. eng., nonlinear dyn. eng. syst., 338, 1651, 533-546, (1992) · Zbl 0748.70012 [20] Leine, R.I.; Van Campen, D.H., Discontinuous bifurcations of periodic solutions, Math. comput. modell., 36, 250-273, (2002) · Zbl 1046.34016 [21] Menon S, Luo ACJ. An analytical prediction of the global period-1 motion in a periodically forced, piecewise linear system. Int J Bifurcat Chaos, in press [22] Luo, A.C.J.; Monen, S., Global chaos in a periodically forced, linear system with a dead-zone restoring force, Chaos, solitons & fractals, 19, 1189-1199, (2004) · Zbl 1069.37027 [23] di Bernardo, M.; Budd, C.J.; Champney, A.R., Normal formal maps for grazing bifurcation in n-dimensional piecewise-smooth dynamical systems, Physica D, 160, 222-254, (2001) · Zbl 1067.37063 [24] di Barnardo, M.; Kowalczyk, K.; Nordmark, A., Bifurcations of dynamical systems with sliding; derivation of normal formal mappings, Physica D, 170, 175-205, (2002) · Zbl 1008.37029 [25] Kunze M. Non-smooth dynamical systems, Lecture Notes in Mathematics 1744. Berlin: Springer; 2000 · Zbl 0965.34026 [26] Popp, K., Non-smooth mechanical systems, J. appl. math. mech., 64, 765-772, (2000) · Zbl 0982.70016
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