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Global invariant manifolds delineating transition and escape dynamics in dissipative systems: an application to snap-through buckling

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Abstract

Invariant manifolds play an important role in organizing global dynamical behaviors. For example, it is found that in multi-well conservative systems where the potential energy wells are connected by index-1 saddles, the motion between potential wells is governed by the invariant manifolds of a periodic orbit around the saddle. In two-degree-of-freedom systems, such invariant manifolds appear as cylindrical conduits which are referred to as transition tubes. In this study, we apply the concept of invariant manifolds to study the transition between potential wells in not only conservative systems, but more realistic dissipative systems, by solving respective proper boundary value problems. The example system considered is a two-mode model of the snap-through buckling of a shallow arch. We define the transition region, \(\mathcal {T}_h\), as a set of initial conditions of a given initial Hamiltonian energy h with which the trajectories can escape from one potential well to another, which in the example system corresponds to snap-through buckling of a structure. The numerical results reveal that in the conservative system, the boundary of the transition region, \(\partial \mathcal {T}_h\), is a cylinder, while in the dissipative system, \(\partial \mathcal {T}_h\) is an ellipsoid. The algorithms developed in the current research from the perspective of invariant manifold provide a robust theoretical–computational framework to study escape and transition dynamics.

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Acknowledgements

The authors would like to thank Mingwu Li for the discussion on COCO and Hinke M. Osinga for stimulating discussion during the nascent stage of this work. We also thank Harry Dankowicz and Jan Sieber for hosting “Advanced Summer School on Continuation Methods for Nonlinear Problems” at UIUC in 2018 from which the authors got to know COCO.

Funding

This work was supported in part by the National Science Foundation under awards 1537349 and 1821145.

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Authors and Affiliations

  1. School of Engineering, Brown University, Providence, RI, 02912, USA

    Jun Zhong

  2. Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    Shane D. Ross

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  1. Jun Zhong
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Correspondence to Jun Zhong.

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Appendices

Appendices

Quadratic equation for the transition ellipsoid

The form of the transition ellipsoid in (52) that mediates the transition in the dissipative system for the snap-through buckling of a shallow arch can be rewritten by the following form:

$$\begin{aligned} a_{\bar{p}_2} \left( \bar{p}_2^0 \right) ^2 + b_{\bar{p}_2} \bar{p}_2^0 + c_{\bar{p}_2}=0, \end{aligned}$$
(69)

where \(a_{\bar{p}_2}\), \(b_{\bar{p}_2}\), and \(c_{\bar{p}_2}\) are given by

$$\begin{aligned}&a_{\bar{p}_2}= \frac{s_2^2}{2 \omega _p(c_x + \omega _p^2)^2}, \\&b_{\bar{p}_2}=\frac{\lambda s_2^2 (1+k_p) (c_x - \lambda ^2) [\bar{q}_2^0 - \bar{q}_1^0 (c_x + \omega _p^2)]}{\omega _p (k_p - 1) (c_x + \omega _p^2)^2 (\lambda ^2 + \omega _p^2)},\\&c_{\bar{p}_2}= c_p - \frac{\lambda ^2 s_2^2 (1+k_p)^2 (c_x - c_y)[\bar{q}_2^0 - \bar{q}_1^0 (c_x + \omega _p^2)]^2}{2 \omega _p (k_p-1)^2 (c_x + \omega _p^2)^2(\lambda ^2 + \omega _p^2)},\\&c_p = \left( \sum \limits _{i=1}^{4} c_p^{(i)}\right) / \left[ 2 \omega _p \left( k_p-1 \right) ^2 \left( \lambda ^2 + \omega _p^2 \right) ^2 \right] - h,\\&c_p^{(1)}=2 k_p s_1^2 \lambda \omega _p \left[ \bar{q}_2- \bar{q}_1 \left( c_x + \omega _p^2 \right) \right] ^2, \\&c_p^{(2)}= 8 k_p s_2^2 \lambda ^2 \omega _p^2 \bar{q}_1 \left( c_x \bar{q}_1 - \bar{q}_2 \right) ,\\&c_p^{(3)}=s_2^2 \lambda ^2 \left( 1+k_p \right) ^2 \left[ \left( c_x \bar{q}_1- \bar{q}_2 \right) ^2+ \bar{q}_1^2\omega _p^4 \right] ,\\&c_p^{(4)}=s_2^2 \omega _p^2 \left( k_p -1 \right) ^2 \left[ \left( c_x \bar{q}_1 - \bar{q}_2) \right) ^2+ \bar{q}_1^2 \lambda ^4 \right] . \end{aligned}$$

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Zhong, J., Ross, S.D. Global invariant manifolds delineating transition and escape dynamics in dissipative systems: an application to snap-through buckling. Nonlinear Dyn 104, 3109–3137 (2021). https://doi.org/10.1007/s11071-021-06509-w

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