Figure 1.
Hydraulic cylinder
-
Previous Article
Existence of periodic solutions of dynamic equations on time scales by averaging
- DCDS-S Home
- This Issue
-
Next Article
Stability analysis of a model on varying domain with the Robin boundary condition
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 & 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. |
| [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. 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. |
| [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. |
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. 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. |
| [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
| [1] |
Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021151 |
| [2] |
Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 |
| [3] |
Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108 |
| [4] |
Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161 |
| [5] |
Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553 |
| [6] |
Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 |
| [7] |
Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137 |
| [8] |
Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007 |
| [9] |
Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 |
| [10] |
Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 |
| [11] |
Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 |
| [12] |
Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 |
| [13] |
Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050 |
| [14] |
Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 |
| [15] |
R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 |
| [16] |
Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347 |
| [17] |
A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373 |
| [18] |
Ling Yun Wang, Wei Hua Gui, Kok Lay Teo, Ryan Loxton, Chun Hua Yang. Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications. Journal of Industrial & Management Optimization, 2009, 5 (4) : 705-718. doi: 10.3934/jimo.2009.5.705 |
| [19] |
Tomás Caraballo, Gábor Kiss. Attractors for differential equations with multiple variable delays. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1365-1374. doi: 10.3934/dcds.2013.33.1365 |
| [20] |
Prashant Shekhar, Abani Patra. Hierarchical approximations for data reduction and learning at multiple scales. Foundations of Data Science, 2020, 2 (2) : 123-154. doi: 10.3934/fods.2020008 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]