# Transient and chaotic low-energy transfers in a system with bistable nonlinearity

## Abstract

The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The mechanisms leading to intense energy exchanges between the linear oscillator, excited by a low-energy impulse, and the nonlinear attachment are addressed. For lightly damped systems, it is shown that two main mechanisms arise: Aperiodic alternating in-well and cross-well oscillations of the nonlinear attachment, and secondary nonlinear beats occurring once the dynamics evolves solely in-well. The description of the former dissipative phenomenon is provided in a two-dimensional projection of the phase space, where transitions between in-well and cross-well oscillations are associated with sequences of crossings across a pseudo-separatrix. Whereas the second mechanism is described in terms of secondary limiting phase trajectories of the nonlinear attachment under certain resonance conditions. The analytical treatment of the two aformentioned low-energy transfer mechanisms relies on the reduction of the nonlinear dynamics and consequent analysis of the reduced dynamics by asymptotic techniques. Direct numerical simulations fully validate our analytical predictions.

- Authors:

- Department of Structural and Geotechnical Engineering, SAPIENZA University of Rome, Rome (Italy)
- Institute of Chemical Physics, RAS, Moscow (Russian Federation)
- College of Engineering, University of Illinois at Urbana–Champaign, Champaign, Illinois 61820 (United States)

- Publication Date:

- OSTI Identifier:
- 22402557

- Resource Type:
- Journal Article

- Journal Name:
- Chaos (Woodbury, N. Y.)

- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1054-1500

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANALYTIC FUNCTIONS; ASYMPTOTIC SOLUTIONS; CHAOS THEORY; COMPUTERIZED SIMULATION; ENERGY TRANSFER; MASS; NONLINEAR PROBLEMS; OSCILLATIONS; OSCILLATORS; PHASE SPACE; TRANSIENTS; TWO-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS

### Citation Formats

```
Romeo, F., E-mail: francesco.romeo@uniroma1.it, Manevitch, L. I., Bergman, L. A., and Vakakis, A.
```*Transient and chaotic low-energy transfers in a system with bistable nonlinearity*. United States: N. p., 2015.
Web. doi:10.1063/1.4921193.

```
Romeo, F., E-mail: francesco.romeo@uniroma1.it, Manevitch, L. I., Bergman, L. A., & Vakakis, A.
```*Transient and chaotic low-energy transfers in a system with bistable nonlinearity*. United States. doi:10.1063/1.4921193.

```
Romeo, F., E-mail: francesco.romeo@uniroma1.it, Manevitch, L. I., Bergman, L. A., and Vakakis, A. Fri .
"Transient and chaotic low-energy transfers in a system with bistable nonlinearity". United States. doi:10.1063/1.4921193.
```

```
@article{osti_22402557,
```

title = {Transient and chaotic low-energy transfers in a system with bistable nonlinearity},

author = {Romeo, F., E-mail: francesco.romeo@uniroma1.it and Manevitch, L. I. and Bergman, L. A. and Vakakis, A.},

abstractNote = {The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The mechanisms leading to intense energy exchanges between the linear oscillator, excited by a low-energy impulse, and the nonlinear attachment are addressed. For lightly damped systems, it is shown that two main mechanisms arise: Aperiodic alternating in-well and cross-well oscillations of the nonlinear attachment, and secondary nonlinear beats occurring once the dynamics evolves solely in-well. The description of the former dissipative phenomenon is provided in a two-dimensional projection of the phase space, where transitions between in-well and cross-well oscillations are associated with sequences of crossings across a pseudo-separatrix. Whereas the second mechanism is described in terms of secondary limiting phase trajectories of the nonlinear attachment under certain resonance conditions. The analytical treatment of the two aformentioned low-energy transfer mechanisms relies on the reduction of the nonlinear dynamics and consequent analysis of the reduced dynamics by asymptotic techniques. Direct numerical simulations fully validate our analytical predictions.},

doi = {10.1063/1.4921193},

journal = {Chaos (Woodbury, N. Y.)},

issn = {1054-1500},

number = 5,

volume = 25,

place = {United States},

year = {2015},

month = {5}

}