22 January 2001 Physics Letters A 279 (2001) 38–46 www.elsevier.nl/locate/pla Role of nonlinear dissipation in the suppression of chaotic escape from a potential well R. Chacón a,∗ , F. Balibrea b , M.A. López c a Departamento de Electrónica e Ingeniería Electromecánica, Universidad de Extremadura, Apartado Postal 382, E-06071 Badajoz, Spain b Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, Campus de Espinardo, E-30100 Murcia, Spain c Departamento de Matemáticas, Escuela Universitaria Politécnica, Universidad de Castilla-La Mancha, E-16071 Cuenca, Spain Received 17 April 2000; received in revised form 12 October 2000; accepted 13 October 2000 Communicated by A.P. Fordy Abstract The inhibition of chaotic escape from a universal escape oscillator due to a periodic parametric perturbation of the quadratic potential term is studied theoretically by means of Poincaré–Melnikov–Arnold analysis, and the predictions are tested against numerical simulations based on a high-resolution grid of initial conditions. It is shown that chaotic escape suppression is impossible under period-1 and period-2 parametric perturbations. The role of a nonlinear damping term, proportional to the nth power of the velocity, on the inhibition scenario is also discussed.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.45.-a 1. Introduction Very diverse oscillatory phenomena present the common characteristic of an incidental escape from a potential well — the subsequent motion being un- bounded. The main properties of such escape phenom- ena can be modeled by a simple oscillator model with a quadratic nonlinearity: (1) ¨ x + x − x 2 =−d(x, ˙ x) + γ cos(ωt), where −d(x, ˙ x) is a general damping force and γ,ω are the forcing amplitude and frequency, respectively. Model (1) has been extensively studied [1–5]. In * Corresponding author. E-mail address: rchacon@unex.es (R. Chacón). particular, the erosion of the nonescaping basin was considered by introducing a linear damping term [4] and a nonlinear damping term proportional to the nth power of the velocity [5]. The conclusion was that, once the parameters of the system are fixed, increasing the power of the nonlinear term has similar effects (i.e., in destroying the safe basins and increasing the basin erosion patterns) to those of decreasing the damping coefficient in the linear damping case. The application of a parametric modulation (PM) of the linear potential term inhibits the chaotic escape when certain resonance conditions are met [6]. For a linear damping, such a PM of the linear potential term (of the normalized model (1)) suppresses chaotic escape more efficiently than for a nonlinear damping of the form −d(x, ˙ x) ≡−(δ 2 x 2 + δ 3 x 4 ) ˙ x , where δ 2,3 are damping coefficients [6]. 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0375-9601(00)00675-7