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

* Corresponding author: Jinli Xu

Received  August 2016 Revised  January 2017 Published  June 2017

Fund Project: The first author is supported by National Nature Science Foundation of China No.11501091, the Fundamental Research Funds for the Central Universities No. 2572017CB22, Postdoctoral Science Foundation of China No.2015M571382 and No.2016T90266, and the Heilongjiang Provincial Natural No.A201401.

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.

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
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.

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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.

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Y. GuoW. 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.

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Z. JiangW. 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.

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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.  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.

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. GuoW. 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. JiangW. 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 1.  Hydraulic cylinder
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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