Figure 1.
Hydraulic cylinder
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October
2017, 10(5): 943-958.
doi: 10.3934/dcdss.2017049
Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors
| 1. | Department of Mathematics, Northeast Forestry University, Harbin, 150040, China |
| 2. | College of Electromechanical Engineering, Northeast Forestry University, Harbin, 150040, China |
In this paper, we study dynamics in delayed nonlinear hydraulic cylinder equation, with particular attention focused on several types of bifurcations. Firstly, basing on a series of original equations, we model a nonlinear delayed differential equations associated with hydraulic cylinder in glue dosing processes for particleboard. Secondly, we identify the critical values for fixed point, Hopf, Hopf-zero, double Hopf and tri-Hopf bifurcations using the method of bifurcation analysis. Thirdly, by applying the multiple time scales method, the normal form near the Hopf-zero bifurcation critical points is derived. Finally, two examples are presented to demonstrate the application of the theoretical results.
Keywords:
Mathematical modeling about hydraulic cylinder,
delayed differential equations,
bifurcation,
multiple time scales,
normal form.
Mathematics Subject Classification:
Primary: 37G10, 37G05; Secondary: 34K27.
Citation:
Yuting Ding, Jinli Xu, Jun Cao, Dongyan Zhang. Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors. Discrete and Continuous Dynamical Systems - S,
2017, 10
(5)
: 943-958.
doi: 10.3934/dcdss.2017049
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References:
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Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Math. Comput. Simul., 122 (2016), 35-54.
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N. Manring,
Hydraulic Control Systems, Wiley-Interscience, New York, 2005. |
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H. E. Merritt,
Hydraulic Control Systems, Wiley-Interscience, New York, 1991. |
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A. H. Nayfeh,
Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981. |
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A. H. Nayfeh,
Order reduction of retarded nonlinear systems--the method of multiple scales versus center-manifold reduction, Nonlinear Dyn., 51 (2008), 483-500.
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H. Wang and W. Jiang,
Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 368 (2010), 9-18.
doi: 10.1016/j.jmaa.2010.03.012. |
show all references
References:
| [1] |
S. A. Campbell and Y. Yuan,
Zero singularities of codimension two and three in delay differential equations, Nonlinearity, 21 (2008), 2671-2691.
doi: 10.1088/0951-7715/21/11/010. |
| [2] |
Y. Choi and V. G. LeBlanc,
Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction, J. Differ. Equ., 227 (2006), 166-203.
doi: 10.1016/j.jde.2005.12.003. |
| [3] |
S. L. Das and A. Chatterjee,
Multiple scales via Galerkin projections: Approximate asymptotics for strongly nonlinear oscillations, Nonlinear Dyn., 32 (2003), 161-186.
doi: 10.1023/A:1024447407071. |
| [4] |
Y. Guo, W. Jiang and B. Niu,
Bifurcation analysis in the control of chaos by extended delay feedback, J. Franklin Inst., 350 (2013), 155-170.
doi: 10.1016/j.jfranklin.2012.10.009. |
| [5] |
Z. Jiang, W. Ma and J. Wei,
Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Math. Comput. Simul., 122 (2016), 35-54.
doi: 10.1016/j.matcom.2015.11.002. |
| [6] |
N. Manring,
Hydraulic Control Systems, Wiley-Interscience, New York, 2005. |
| [7] |
H. E. Merritt,
Hydraulic Control Systems, Wiley-Interscience, New York, 1991. |
| [8] |
A. H. Nayfeh,
Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981. |
| [9] |
A. H. Nayfeh,
Order reduction of retarded nonlinear systems--the method of multiple scales versus center-manifold reduction, Nonlinear Dyn., 51 (2008), 483-500.
doi: 10.1007/s11071-007-9237-y. |
| [10] |
H. Wang and W. Jiang,
Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 368 (2010), 9-18.
doi: 10.1016/j.jmaa.2010.03.012. |
Figure 2.
Principle of hydraulic drive unit
Figure 3.
Simulated solution of system (5) associated with the time history for $\tau=0$ , showing a stable equilibrium
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