In this paper, periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with two vector fields are discussed. Constructing periodic motions and homoclinic orbits in discontinuous dynamical systems is very significant in mathematics and engineering applications, and how to construct periodic motions and homoclinic orbits is a central issue in discontinuous dynamical systems. Herein, how to construct periodic motions and homoclinic orbits is presented through studying a simple discontinuous dynamical system on a domain confined by two prescribed energies. The simple discontinuous dynamical system has energy-increasing and energy-decreasing vector fields. Based on the two vector fields and the corresponding switching rules, periodic motions and homoclinic orbits in such a simple discontinuous dynamical system are studied. The analytical conditions of bouncing, grazing, and sliding motions at the two energy boundaries are presented first. Periodic motions and homoclinic orbits in such a discontinuous dynamical system are determined through the specific mapping structures, and the corresponding stability is also presented. Numerical illustrations of periodic motions and homoclinic orbits are given for constructed complex motions. Through this study, using discontinuous dynamical systems, one can construct specific complex motions for engineering applications, and the corresponding mathematical methods and computational strategies can be developed.
REFERENCES
1.
J. P. D.
Hartog
, “Forced vibrations with combined Coulomb and viscous friction
,” Trans. Am. Soc. Mech. Eng.
53
, 107
–115
(1931
).2.
A. F.
Filippov
, “Differential equations with discontinuous right-hand side
,” Am. Math. Soc. Trans. Ser. 2
42
, 199
–231
(1964
). 3.
A. F.
Filippov
, “Classical solutions of differential equations with multi-valued right-hand side
,” SIAM J. Control
5
(4
), 609
–621
(1967
). 4.
A. F.
Filippov
, Differential Equations with Discontinuous with Right-Hand Sides
(Kluwer Academic
, 1988
).5.
M. A.
Aizerman
and E. S.
Pyatnitsky
, “Fundamentals of the theory of discontinuous systems
,” Autom. Remote Control Ser. 1
35
, 1066
–1079
(1974
).6.
M. A.
Aizerman
and E. S.
Pyatnitsky
, “Fundamentals of the theory of discontinuous systems
,” Autom. Remote Control Ser. 2
35
, 1241
–1262
(1974
).7.
V.
Utkin
, “Variable structure systems with sliding modes
,” IEEE Trans. Autom. Control
22
(2
), 212
–222
(1977
). 8.
S. F.
Masri
and T. K.
Caughey
, “On the stability of the impact damper
,” ASME J. Appl. Mech.
33
, 586
–592
(1966
). 9.
S. F.
Masri
, “General motion of impact dampers
,” J. Acoust. Soc. Am.
47
(1B
), 229
–237
(1970
). 10.
M. I.
Feigin
, “Doubling of the oscillation period with C-bifurcations in piecewise-continuous systems
,” J. Appl. Math. Mech.
34
, 822
–830
(1970
). 11.
M. I.
Feigin
, “On the structure of C-bifurcation boundaries of piecewise-continuous systems
,” J. Appl. Math. Mech.
42
, 885
–895
(1978
). 12.
S. W.
Shaw
and P. J.
Holmes
, “A periodically forced piecewise linear oscillator
,” J. Sound Vib.
90
(1
), 129
–155
(1983
). 13.
D.
Karnopp
, “Computer simulation of stick-slip friction in mechanical dynamic systems
,” J. Dyn. Syst. Meas. Control
107
, 100
–103
(1985
). 14.
R. I.
Leine
, A.
de Kraker
, D. H.
van Campen
, and L.
van den Steen
, “Stick-slip vibrations induced by alternate friction models
,” Nonlinear Dyn.
16
(1
), 41
–54
(1998
). 15.
A. B.
Nordmark
, “Non-periodic motion caused by grazing incidence in an impact oscillator
,” J. Sound Vib.
145
(2
), 279
–297
(1991
). 16.
A. B.
Nordmark
, “Universal limit mapping in grazing bifurcations
,” Phys. Rev. E
55
, 266
–270
(1997
). 17.
A. B.
Nordmark
, “Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators
,” Nonlinearity
14
, 1517
–1542
(2001
). 18.
A. B.
Nordmark
and P.
Kowalczyk
, “A codimension-two scenario of sliding solutions in grazing-sliding bifurcations
,” Nonlinearity
19
, 1
–26
(2006
). 19.
R. I.
Leine
and D. H.
Van Campen
, “Discontinuous bifurcations of periodic solutions
,” Math. Comput. Model.
36
(3
), 259
–273
(2002
). 20.
P.
Kowalczyk
and P. T.
Piiroinen
, “Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator
,” Physica D
237
(8
), 1053
–1073
(2008
). 21.
V. A.
Bazhenov
, P. P.
Lizunov
, O. S.
Pogorelova
, T. G.
Postnikova
, and V. V.
Otrashevskaia
, “Stability and bifurcations analysis for 2-DOF vibro-impact system by parameter continuation method: Part I: Loading curve
,” J. Appl. Nonlinear Dyn.
4
(4
), 357
–370
(2015
). 22.
V. V.
Bazhenov
, O. S.
Pogorelova
, and T. G.
Postnikova
, “Breakup of closed curve—Quasiperiodic route to chaos in vibro-impact system
,” Discontinuity Nonlinearity Complexity
8
(3
), 299
–311
(2019
). 23.
M. U.
Akhmet
and A.
Kıvılcım
, “Van der Pol oscillators generated from grazing dynamics
,” Discontinuity Nonlinearity Complexity
7
(3
), 259
–274
(2018
). 24.
A. C. J.
Luo
, Singularity and Dynamics on Discontinuous Vector Fields
(Elsevier
, Amsterdam
, 2006
).25.
A. C. J.
Luo
and L. D.
Chen
, “Periodic motion and grazing in a harmonically forced, piecewise, linear oscillator with impacts
,” Chaos Solitons Fractals
24
, 567
–578
(2005
). 26.
A. C. J.
Luo
and B. C.
Gegg
, “Stick and non-stick periodic motions in a periodically forced oscillator with dry-friction
,” J. Sound Vib.
291
, 132
–168
(2006
). 27.
A. C. J.
Luo
and B. C.
Gegg
, “On the mechanism of stick and non-stick, periodic motions in a forced linear oscillator including dry friction
,” ASME J. Vib. Acoust.
128
, 97
–105
(2006
). 28.
A. C. J.
Luo
, “A theory for flow switchability in discontinuous dynamical systems
,” Nonlinear Anal. Hybrid Syst.
2
, 1030
–1061
(2008
). 29.
A. C. J.
Luo
, “On flow barriers and switchability in discontinuous dynamical systems
,” Int. J. Bifurcation Chaos
21
(1
), 1
–76
(2011
). 30.
31.
J. Z.
Huang
and A. C. J.
Luo
, “Complex dynamics of bouncing motions at boundaries and corners in a discontinuous dynamical system
,” ASME J. Comput. Nonlinear Dyn.
12
, 061014
(2017
). 32.
X.
Tang
, X.
Fu
, and X.
Sun
, “Periodic motion for an oblique impact system with single degree of freedom
,” J. Vib. Test. Syst. Dyn.
3
(3
), 71
–89
(2019
). 33.
X.
Tang
, X.
Fu
, and X.
Sun
, “The dynamical behavior of a two degrees of freedom oblique impact system
,” Discontinuity Nonlinearity Complexity
9
(1
), 117
–139
(2020
). 34.
L.
Li
and A. C. J.
Luo
, “Periodic orbits in a second-order discontinuous system with an elliptic boundary
,” Int. J. Bifurcation Chaos
26
(13
), 1650224
(2016
). 35.
S.
Guo
and A. C. J.
Luo
, “An analytical prediction of periodic motions in a discontinuous dynamical system
,” J. Vib. Test. Syst. Dyn.
4
(4
), 377
–388
(2020
). 36.
S.
Guo
and A. C. J.
Luo
, “A parameter study on periodic motions in a discontinuous dynamical system with two circular boundaries
,” Discontinuity Nonlinearity Complexity
10
, 289
–309
(2021
). 37.
S.
Guo
and A. C. J.
Luo
, “On existence and bifurcations of periodic motions in discontinuous dynamical systems
,” Int. J. Bifurcation Chaos
31
(4
), 2150063
(2021
). 38.
P.
Feketa
, V.
Klinshov
, and L.
Lucken
, “A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly
,” Commun. Nonlinear Sci. Numer. Simul.
103
, 105955
(2021
). 39.
N.
Levinson
, “A second order differential equation with singular solutions
,” Ann. Math.
50
, 127
–154
(1949
). 40.
V.
Belykh
, N.
Barabash
, and I.
Belykh
, “A Lorenz-type attractor in a piecewise-smooth system: Rigorous results
,” Chaos
29
, 103108
(2019
). 41.
V.
Belykh
, N.
Barabash
, and I.
Belykh
, “Bifurcations of chaotic attractors in a piecewise-smooth Lorenz-type system
,” Autom. Remote Control
81
(8
), 1385
–1393
(2020
). 42.
S.
Guo
and A. C. J.
Luo
, “Constructed limit cycles in a discontinuous dynamical system with multiple vector fields
,” J. Vib. Test. Syst. Dyn.
5
, 33
–51
(2021
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
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