Eur. Phys. J. Plus (2017) 132: 116
https://doi.org/10.1140/epjp/i2017-11352-9

Regular Article

Nonlinear dynamics behavior analysis of the spatial configuration of a tendril-bearing plant

¹ Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System, School of Mechanical Engineering, Tianjin University of Technology, Tianjin, China
² Tianjin Key Laboratory of Nonlinear Dynamics and Control Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin, China

^* e-mail: wangweifrancis@tju.edu.cn

Received: 22 November 2016
Accepted: 14 January 2017
Published online: 7 March 2017

Abstract

Tendril-bearing plants appear to have a spiraling shape when tendrils climb along a support during growth. The growth characteristics of a tendril-bearer can be simplified to a model of a thin elastic rod with a cylindrical constraint. In this paper, the connection between some typical configuration characteristics of tendrils and complex nonlinear dynamic behavior are qualitatively analyzed. The space configuration problem of tendrils can be explained through the study of the nonlinear dynamic behavior of the thin elastic rod system equation. In this study, the complex non-Z₂ symmetric critical orbits in the system equation under critical parameters were presented. A new function transformation method that can effectively maintain the critical orbit properties was proposed, and a new nonlinear differential equations system containing complex nonlinear terms can been obtained to describe the cross section position and direction of a rod during climbing. Numerical simulation revealed that the new system can describe the configuration of a rod with reasonable accuracy. To adequately explain the growing regulation of the rod shape, the critical orbit and configuration of rod are connected in a direct way. The high precision analytical expressions of these complex non-Z₂ symmetric critical orbits are obtained by introducing a suitable analytical method, and then these expressions are used to draw the corresponding three-dimensional configuration figures of an elastic thin rod. Combined with actual tendrils on a live plant, the space configuration of the winding knots of tendril is explained by the concept of heteroclinic orbit from the perspective of nonlinear dynamics, and correctness of the theoretical analysis was verified. This theoretical analysis method could also be effectively applied to other similar slender structures.