July  2011, 16(1): 319-332. doi: 10.3934/dcdsb.2011.16.319

Vanishing singularity in hard impacting systems

1. 

Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, United States

2. 

Department of Physical Sciences, Indian Institute of Science Education & Research, Mohanpur-741252, Nadia, West Bengal, India

3. 

School of Electrical, Electronic and Computer Engineering, Newcastle University, NE1 7RU, England, United Kingdom

Received  February 2010 Revised  June 2010 Published  April 2011

It is known that the Jacobian of the discrete-time map of an impact oscillator in the neighborhood of a grazing orbit depends on the square-root of the distance the mass would have gone beyond the position of the wall if the wall were not there. This results in an infinite stretching of the phase space, known as the square-root singularity. In this paper we look closer into the Jacobian matrix and find out the behavior of its two parameters---the trace and the determinant, across the grazing event. We show that the determinant of the matrix remains invariant in the neighborhood of a grazing orbit, and that the singularity appears only in the trace of the matrix. Investigating the character of the trace, we show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency.
Citation: Soumya Kundu, Soumitro Banerjee, Damian Giaouris. Vanishing singularity in hard impacting systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 319-332. doi: 10.3934/dcdsb.2011.16.319
References:
[1]

Journal of Sound & Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8.  Google Scholar

[2]

Journal of Sound and Vibration, 154 (1992), 95-115. doi: 10.1016/0022-460X(92)90406-N.  Google Scholar

[3]

Chaos, Solitons & Fractals, 9 (1998), 1439-1443. doi: 10.1016/S0960-0779(98)00164-7.  Google Scholar

[4]

Nonlinear Dynamics, 15 (1998), 391-409. doi: 10.1023/A:1008209513877.  Google Scholar

[5]

Journal of Sound & Vibration, 228 (1999), 243-264. doi: 10.1006/jsvi.1999.2318.  Google Scholar

[6]

Nonlinear Dynamics, 50 (2007), 701-716. doi: 10.1007/s11071-006-9180-3.  Google Scholar

[7]

In "International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series," 96 (2007), 012119. Google Scholar

[8]

Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.  Google Scholar

[9]

Phys. Rev. E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.  Google Scholar

[10]

Springer Verlag (Applied Mathematical Sciences), London, 2008. Google Scholar

[11]

Physics Letters A, 354 (2006), 281-287. doi: 10.1016/j.physleta.2006.01.025.  Google Scholar

[12]

International Journal of Nonlinear Mechanics, 43 (2008), 504-513. doi: 10.1016/j.ijnonlinmec.2008.04.001.  Google Scholar

[13]

Philosophical Transactions of the Royal Society of London, Part A, 366 (2008), 679-704. Google Scholar

[14]

Springer Verlag, Berlin, 2004. Google Scholar

[15]

Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.  Google Scholar

[16]

IEEE Transactions on Circuits and Systems-I, 47 (2000), 633-643. doi: 10.1109/81.847870.  Google Scholar

[17]

Int. J. Robust & Nonlinear Control, 17 (2007), 1405-1429. doi: 10.1002/rnc.1252.  Google Scholar

show all references

References:
[1]

Journal of Sound & Vibration, 90 (1983), 129-155. doi: 10.1016/0022-460X(83)90407-8.  Google Scholar

[2]

Journal of Sound and Vibration, 154 (1992), 95-115. doi: 10.1016/0022-460X(92)90406-N.  Google Scholar

[3]

Chaos, Solitons & Fractals, 9 (1998), 1439-1443. doi: 10.1016/S0960-0779(98)00164-7.  Google Scholar

[4]

Nonlinear Dynamics, 15 (1998), 391-409. doi: 10.1023/A:1008209513877.  Google Scholar

[5]

Journal of Sound & Vibration, 228 (1999), 243-264. doi: 10.1006/jsvi.1999.2318.  Google Scholar

[6]

Nonlinear Dynamics, 50 (2007), 701-716. doi: 10.1007/s11071-006-9180-3.  Google Scholar

[7]

In "International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series," 96 (2007), 012119. Google Scholar

[8]

Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.  Google Scholar

[9]

Phys. Rev. E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.  Google Scholar

[10]

Springer Verlag (Applied Mathematical Sciences), London, 2008. Google Scholar

[11]

Physics Letters A, 354 (2006), 281-287. doi: 10.1016/j.physleta.2006.01.025.  Google Scholar

[12]

International Journal of Nonlinear Mechanics, 43 (2008), 504-513. doi: 10.1016/j.ijnonlinmec.2008.04.001.  Google Scholar

[13]

Philosophical Transactions of the Royal Society of London, Part A, 366 (2008), 679-704. Google Scholar

[14]

Springer Verlag, Berlin, 2004. Google Scholar

[15]

Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.  Google Scholar

[16]

IEEE Transactions on Circuits and Systems-I, 47 (2000), 633-643. doi: 10.1109/81.847870.  Google Scholar

[17]

Int. J. Robust & Nonlinear Control, 17 (2007), 1405-1429. doi: 10.1002/rnc.1252.  Google Scholar

[1]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021096

[2]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (II): The sum of indices of equilibria is $ 1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021101

[3]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[4]

Bruce Kitchens, Brian Marcus, Benjamin Weiss. Roy Adler and the lasting impact of his work. Journal of Modern Dynamics, 2018, 13: v-x. doi: 10.3934/jmd.2018v

[5]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

[6]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[7]

Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3043-3068. doi: 10.3934/dcdsb.2020219

[8]

Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021084

[9]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[10]

Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021034

[11]

Yusra Bibi Ruhomally, Muhammad Zaid Dauhoo, Laurent Dumas. A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021011

[12]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[13]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089

[14]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023

[15]

Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021117

[16]

Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021047

[17]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[18]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[19]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[20]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (11)

[Back to Top]