Transition to Chaos in the Self-Excited System with a Cubic Double Well Potential and Parametric Forcing

Abstract We examine the Melnikov criterion for a global homoclinic bifurcation and a possible transition to chaos in case of a single degree of freedom nonlinear oscillator with a symmetric double well nonlinear potential. The system was subjected simultaneously to parametric periodic forcing and self-excitation via negative damping term. Detailed numerical studies confirm the analytical predictions and show that transitions from regular to chaotic types of motion are often associated with increasing the energy of an oscillator and its escape from a single well.

[1]  G. Gladwell,et al.  Solid mechanics and its applications , 1990 .

[2]  Stefano Lenci,et al.  Higher-order Melnikov functions for single-dof mechanical oscillators: Theoretical treatment and applications , 2004 .

[3]  Wanda Szempliiqska-Stupnicka.  The Analytical Predictive Criteria for Chaos and Escape in Nonlinear Oscillators: A Survey , 2004 .

[4]  Stefano Lenci,et al.  Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator , 2003 .

[5]  M. Siewe Siewe,et al.  Resonant oscillation and homoclinic bifurcation in a Φ6-Van der Pol oscillator , 2004 .

[6]  Paul Woafo,et al.  Bifurcations and chaos in the triple-well Φ6-Van der Pol oscillator driven by external and parametric excitations , 2005 .

[7]  Reshaping-induced order-chaos routes in a damped driven Helmholtz oscillator , 2005 .

[8]  Giuseppe Rega,et al.  Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator , 1995 .

[9]  Hongjun Cao.  Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators , 2005 .

[10]  Elzbieta Tyrkiel,et al.  On the Role of Chaotic saddles in Generating Chaotic Dynamics in Nonlinear Driven oscillators , 2005, Int. J. Bifurc. Chaos.

[11]  J. Thompson,et al.  Chaotic phenomena triggering the escape from a potential well , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Kazimierz Szabelski.  The vibrations of self-excited system with parametric excitation and non-symmetric elasticity characteristic , 1991 .

[13]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[14]  VIBRATION ANALYSIS OF A SELF-EXCITED SYSTEM WITH PARAMETRIC FORCING AND NONLINEAR STIFFNESS , 1999 .

[15]  Tomasz Kapitaniak,et al.  Chaotic Oscillations in Mechanical Systems , 1991 .

[16]  Stefano Lenci,et al.  A unified control framework of the non-regular dynamics of mechanical oscillators , 2004 .

[17]  M. Lakshmanan,et al.  On the non-integrability of a family of Duffing-van der Pol oscillators , 1993 .

[18]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[19]  Shaopu Yang,et al.  Investigation on chaotic motion in hysteretic non-linear suspension system with multi-frequency excitations , 2004 .

[21]  Prediction of horseshoe chaos in BVP and DVP oscillators , 1992 .

[22]  Mariusz M Holicke,et al.  MELNIKOV'S METHOD AND STICK–SLIP CHAOTIC OSCILLATIONS IN VERY WEAKLY FORCED MECHANICAL SYSTEMS , 1999 .

[23]  Guanrong Chen,et al.  Global and Local Control of homoclinic and heteroclinic bifurcations , 2005, Int. J. Bifurc. Chaos.

[24]  Miguel A. F. Sanjuán,et al.  Analytical Estimates of the Effect of nonlinear damping in some nonlinear oscillators , 2000, Int. J. Bifurc. Chaos.

[25]  W. Szemplinska-Stupnicka,et al.  Bifurcations phenomena in a nonlinear oscillator: approximate analytical studies versus computer simulation results , 1993 .

[26]  Grzegorz Litak,et al.  Suppression of chaos by weak resonant excitations in a non-linear oscillator with a non-symmetric potential , 2004 .

[27]  Paul Woafo,et al.  Linear feedback and parametric controls of vibrations and chaotic escape in a Φ6 potential , 2003 .

[28]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[29]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[30]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[31]  Bernold Fiedler,et al.  Ergodic theory, analysis, and efficient simulation of dynamical systems , 2001 .