Talk:Duffing equation

Phase Space

Please add a plot of the trajectory in the phase space. lbertolotti 8 febb —Preceding unsigned comment added by Lbertolotti (talk • contribs) 23:39, 7 February 2009 (UTC)

Chaotic movement ?

When a physical system can be described by a specific differential equation, then the system ¨follows orders¨. The system executes the differential equation. The word ¨chaotic¨ may not be used. The complex behaviour of the system is just beyond our comprehension. — Preceding unsigned comment added by 178.84.232.96 (talk) 16:36, 25 October 2014 (UTC)

Read more. Highlander45 (talk) 07:22, 21 August 2015 (UTC)

Frequency response by the homotopy analysis method

Some points I would like to point out regarding this section:

• The whole section was written by FaridTF. The only reference given in this section is a paper written by "Farid Tajaddodianfar". It seems as if this section was added to promote his own publication.
• There is no such thing as a Duffing oscillator with quadratic nonlinearity. The Duffing equation is given in the article introduction. There is no argument in adding additional nonlinear restoring force terms to describe pratical problems. In some cases, it is also necessary to add nonlinear damping terms. However, there are simply to many extensions to mention them all.
• Although it might be worth adding information regarding the frequency response, the solution obtained by HAM is exactly the same from a Harmonic Balance approach (see for example: D. W. Jordan and P. Smith. Nonlinear Ordinary Differential Equations. Oxford University Press.). Additional methods of solution are given in the respective section.
• Giving the frequency response without explanation (multiple attractors, stability, hysteresis, ...) makes no sense as an equation is not per se useful information.

I removed the parts which are not relevant for Duffing equation itself (relating to quadratic stiffness terms) and added the expand section template. --77.176.198.129 (talk) 12:13, 12 November 2016 (UTC)

Thank you! That is a very nice reference, the Jordan & Smith book. It has been added to the article. Regards, Crowsnest (talk) 18:10, 12 November 2016 (UTC)

Parameter Definitions

Duffing oscillator plot, containing phase plot, trajectory, strange attractor, Poincare section, and double well potential plot. The parameters are $\alpha =-1$, $\beta =0.25$, $\delta =0.1$, $\gamma =2.5$, and $\omega =2$.

The strange attractor of the Duffing oscillator, through 4 periods ($8\pi$ time). Coloration shows how the points flow. ($\alpha =-1$, $\beta =1$, $\delta =0.02$, $\gamma =3$, and $\omega =1$)

A Poincaré section of the forced Duffing equation suggesting chaotic behaviour ($\alpha =?-1$, $\beta =?5$, $\delta =?0.02$, $\gamma =?8$, and $\omega =?0.5$).

The current article has some issues with parameter definitions -- especially when comparing with the plots. For example, the first figure uses a potential of the form $V(x)=-\alpha x^{2}+\beta x^{4}$ while the definition of the parameters in the main text uses a potential of the form $V(x)=\alpha x^{2}/2+\beta x^{4}/4$ (note both the sign difference and the normalization difference). This is very confusing for readers. I know this is in Start-class, but let me know if you would like me to try correcting these issues when I have time.

Update: Using the formula from the main text

$${\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t),$$

I have verified updated two of the captions. I have no idea how to reproduce the third yet...

Since most of the interesting dynamics (chaos) happens for $\alpha <0$, the dimensional analysis section should really include an additional sign on the linear term and define $\sigma =\omega /{\sqrt {\vert \alpha \vert }}$. (Or one should normalized based on the $y^{3}$ which generally better be non-negative.)

MichaelMcNeilForbes (talk) 04:48, 5 September 2023 (UTC)