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Dynamic properties of VDP-CPG model in rhythmic movement with delay

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Special Issues: Applications of delay differential equations in biology

In this paper, Van Der Pol (VDP) oscillators are used as the output signal of central pattern generator (CPG), and a VDP-CPG network system of quadruped with four primary gaits (walk, trot, pace and bound) is established. The existence conditions of Hopf bifurcations for VDP-CPG systems corresponding to four primary gaits are given, and the coupling strength ranges between oscillators for four gaits are obtained. Numerical simulations are used to support theoretical analysis.
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1. M. Land, Eye movements in man and other animals, Vision Res., 162 (2019), 1-7.

2. M. Manookin, S. Patterson, C. Linehan, Neural mechanisms mediating motion sensitivity in parasol ganglion cells of the primate retina, Neuron, 97 (2018), 1327-1340.e4.

3. M. Creamer, O. Mano, D. A. Clark, Visual control of walking speed in drosophila, Neuron 100 (2018), 1460-1473.e6.

4. T. Marques, M. T. Summers, G. Fioreze, M. Fridman, R. F. Dias, M. B. Feller, et al., A role for mouse primary visual cortex in motion perception, Curr. Biol., 28 (2018), 1703-1713.e6.

5. F. Delcomyn, Neural basis of rhythmic behavior in animals, Science, 210 (1980), 492-498.

6. K. Sigvardt, T. Williams, Models of central pattern generators as oscillators:the lamprey locomotor CPG, in Seminars in Neuroscience, Academic Press, (1992), 37-46.

7. S. Hooper, Central pattern generators, Current Biology, 10 (2000), 176-179.

8. T. Yamaguchi, The central pattern generator for forelimb locomotion in the cat, in Progress in Brain Research, Elsevier, (2004), 115-122.

9. C. Bal, G. O. Koca, D. Korkmaz, Z. H. Akpolat, M. Ay, CPG-based autonomous swimming control for multi-tasks of a biomimetic robotic fish, Ocean Eng., 189 (2019), 106334.

10. D. Tran, L. Koo, Y. Lee, H. Moon, S. Parket, J. C. Koo, et al., Central pattern generator based reflexive control of quadruped walking robots using a recurrent neural network, Rob. Auton. Sys., 62 (2014), 1497-1516.

11. J. Zhang, F. Gao, X. Han, X. Chen, X. Han, Trot gait design and CPG method for a quadruped robot, J. Bionic. Eng., 11 (2014), 18-25.

12. H. Xu, J. Gan, J. Ren, B. R. Wang, Y. L. Jin, Gait CPG adjustment for a quadruped robot based on Hopf oscillator, J. Syst. Simul., 29 (2017), 3092-3099.

13. H. Liu, W. Jia, L. Bi, Hopf oscillator based adaptive locomotion control for a bionic quadruped robot, 2017 IEEE International Conference on Mechatronics and Automation, 2017. Available from: https://ieeexplore.ieee.org/abstract/document/8015944/.

14. C. Liu, Q. Chen, J. Zhang, Coupled Van der Pol oscillators utilised as central pattern generators for quadruped locomotion, 2009 Chinese Control and Decision Conference, 2009. Available from: https://ieeexplore.ieee.org/abstract/document/5192385.

15. S. Dixit, A. Sharma, M. Shrimali, The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling, Phys. Lett. A, 383 (2019), 125930.

16. J. Collins, I. Stewart, Coupled nonlinear oscillators and the symmetries of animal gaits, J. Nonlinear Sci., 3 (1993), 349-392.

17. P. L. Buono, M. Golubitsky, Models of central pattern generators for quadruped locomotion I. Primary gaits, J. Math. Biol., 42 (2001), 291-326.

18. P. L. Buono, Models of central pattern generators for quadruped locomotion II. Secondary gaits, J. Math. Biol., 42 (2001), 327-346.

19. Y. Song, J. Xu, T. Zhang, Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling, CHA, 21 (2011), 023111.

20. C. Zhang, B. Zheng, L. Wang, Multiple Hopf bifurcation of three coupled van der Pol oscillators with delay, Appl. Math. Comput., 217 (2011), 7155-7166.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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