Oscillator with third and fifth order terms?

[HomogenousCow]

We usually only consider the first order term for an oscillation, are there any papers on extending that model and including third and fifth order terms (since only odd power terms would cause a periodic motion)?
The ODE would look like x''=-αx-βx^3+O(x^5)

 

[andrien]

here is a one,but you may not like it
http://link.springer.com/article/10.1007%2FBF01962591?LI=true#page-1

 

[boneh3ad]

Look up the Duffing equation and its solution. Your equation would be a special case of that if only the terms up to O(x³) are considered.

 

[Enthalpy]

Added even terms would let oscillate as well. They are negligible if your system has an odd response, that is, its transfer function is odd, and then the even components of the series expansion are zero.

Many materials behave symmetrically hence build an odd transfer function, but this approximation would be grossly false in an electronic oscillator for instance.

 

[PJC01]

Yes, there have been numerous papers discussing the inclusion of higher order terms in the model for oscillations. These higher order terms can be important in certain systems, such as nonlinear oscillators or systems with strong nonlinearities. Including third and fifth order terms allows for a more accurate representation of the dynamics of the system, especially in cases where the first order term may not be sufficient.

Some papers have also explored the effects of even order terms, which can also contribute to periodic motion in some systems. Additionally, there have been studies on the stability and bifurcations of oscillators with higher order terms, as well as their applications in various fields such as engineering, physics, and biology.

Overall, the inclusion of higher order terms in the model for oscillations is an active area of research and has provided valuable insights into the behavior of complex systems.