On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.

[1]  N. Mclachlan.  Theory and Application of Mathieu Functions , 1965 .

[2]  K. Peleg,et al.  Parameter sensitivity of non-linear vibration systems , 1989 .

[3]  A discussion of an analytical method of controlling chaos in Duffing's oscillator , 1994 .

[4]  Wanda Szemplińska-Stupnicka.  Higher harmonic oscillations in heteronomous non-linear systems with one degree of freedom , 1968 .

[5]  H. Isomäki,et al.  Absence of inversion-symmetric limit cycles of even periods and the chaotic motion of Duffing's oscillator , 1984 .

[6]  On the Third Superharmonic Resonance in the Duffing Oscillator , 1994 .

[7]  A. Hassan.  Use of transformations with the higher order method of multiple scales periodic response of harmonically excited non-linear oscillators. I: Tranformation of derivative , 1994 .

[8]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[9]  A. Hassan.  Use Of Transformations With The Higher Order Method Of Multiple Scales To Determine The Steady State Periodic Response Of Harmonically Excited Non-linear Oscillators, Part I: Transformation Of Derivative , 1994 .

[10]  von Boehm J,et al.  Chaotic motion of a periodically driven particle in an asymmetric potential well. , 1986, Physical review. A, General physics.

[11]  J. Bajkowski,et al.  The 12 subharmonic resonance and its transition to chaotic motion in a non-linear oscillator , 1986 .

[12]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[13]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[14]  W. Szemplińska-Stupnicka,et al.  Chaotic and regular motion in nonlinear vibrating systems , 1988 .

[15]  T. D. Burton,et al.  Extraneous solutions predicted by the harmonic balance method , 1995 .

[16]  Vimal Singh,et al.  Perturbation methods , 1991 .

[17]  S. T. Noah,et al.  A Generalized Hill’s Method for the Stability Analysis of Parametrically Excited Dynamic Systems , 1982 .

[18]  T. D. Burton,et al.  On the Steady State Response and Stability of Non-Linear Oscillators Using Harmonic Balance , 1993 .

[19]  N. Bogolyubov,et al.  Asymptotic Methods in the Theory of Nonlinear Oscillations , 1961 .

[20]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[21]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[22]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[23]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[24]  Wanda Szemplińska-Stupnicka.  Secondary resonances and approximate models of routes to chaotic motion in non-linear oscillators , 1987 .

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