American Institute of Mathematical Sciences

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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] N. Manring, Hydraulic Control Systems, Wiley-Interscience, New York, 2005.Google Scholar [7] H. E. Merritt, Hydraulic Control Systems, Wiley-Interscience, New York, 1991.Google Scholar [8] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] N. Manring, Hydraulic Control Systems, Wiley-Interscience, New York, 2005.Google Scholar [7] H. E. Merritt, Hydraulic Control Systems, Wiley-Interscience, New York, 1991.Google Scholar [8] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981. Google Scholar [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. Google Scholar [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. Google Scholar
Hydraulic cylinder
Principle of hydraulic drive unit
Simulated solution of system (5) associated with the time history for $\tau=0$, showing a stable equilibrium
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