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RESEARCH PAPERS

# On Global Transversality and Chaos in Two-Dimensional Nonlinear Dynamical Systems

[+] Author and Article Information
Albert C. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805aluo@siue.edu

J. Comput. Nonlinear Dynam 2(3), 242-248 (Feb 20, 2007) (7 pages) doi:10.1115/1.2727494 History: Received August 20, 2006; Revised February 20, 2007

## Abstract

In this paper, the global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function ($L$-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are derived. Numerical simulations are carried out for illustrations of the analytical conditions. From analytical and numerical results, the simple zero of the energy increment (or the Melnikov function) may not imply that chaos exists. The conditions for the global transversality and tangency to the separatrix may be independent of the Melnikov function. Therefore, the analytical criteria for chaotic motions in nonlinear dynamical systems need to be further developed. The methodology presented in this paper is applicable to nonlinear dynamical systems without any separatrix.

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## Figures

Figure 1

Separatrix and subdomains of the integrable Duffing oscillator in Eq. 13(α1=α2=1)

Figure 2

Global tangency and global transversality intervals on the separatrix: (a) grazing displacement versus grazing phase, (b) grazing velocity versus grazing phase, (c) G(1) function varying with grazing phase, and (d) inner and outer grazing on the separatrix (α1=α2=1, δ=0.20, Q0=0.3). ∂Ω⃗ii(i=1,2) and ∂Ω⃗33 represent the grazing occurrence at the separatrix in the domain Ωi and Ω3, which are depicted by the dotted and solid curves, respectively. ∂Ω⃖3i(i=1,2) (or ∂Ω⃗3i) represents the flow globally transversal to the separatrix from domain Ωi to Ω3 (or Ω3 to Ωi).

Figure 3

Bifurcation scenario for the excitation amplitude: (a) vector amplitude and (b) L function of Poincaré mapping points (α1=α2=1, δ=0.4, and Ω=1). LP, GP, and GC represent the local periodic motion, global periodic motion, and global chaotic motion, respectively. LP-i is the local motion in domain Ωi.

Figure 4

A period-3 motion: (a) phase plane, (b) G function, (c) displacement, and (d) L function (α1=α2=1, δ=0.4, Q0=0.46, and Ω=1). The initial condition is x0=0.5723444829, y0=0.5820480550 for t0=0. In (a) and (b), the circular symbols represent the global transversal points at the separatrix. In (c) and (d), the circular symbols are the periodic points of the period-3 motion. The separatrix in phase plane is represented by a thin dashed curve.

Figure 5

A period-1 global grazing motion: (a) phase plane, (b) G function, (c) G(1) function, and (d) L function (α1=α2=1, δ=0.4, Q0=0.255, and Ω=1). The initial condition is x0=0.5962372861, y0=0.5348290607 (right) and x0=−0.6231881765, y0=0.4353772424 (left) for t0=0. In (a)–(c), the circular symbols represent the global tangential points at the separatrix. In (d), the circular symbols are the periodic points of periodic motion. Solid and dashed curves represent the periodic responses in right and left sides of the potential wells. The separatrix in the phase plane is represented by a thin dashed curve.

Figure 6

Chaotic motion: (a) Poincaré mapping section and (b) L function of Poincaré mapping points (α1=α2=1, δ=0.4, Q0=0.4, and Ω=1). The initial condition is x0=−0.5133980853, y0=0.6237605998 for t0=0. The separatrix in the Poincaré mapping section is represented by a thin dashed curve.

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