Consider a mass $m$ attached to a linear dashpot with constant $c$ and a nonlinear spring that exhibits the following response $F=x^3-x($recall the spring force resists motion$).$ The mass is driven by a sinusoidal driver $F(t)=cos(t),$ where $$ is the driving frequency.

Describe the qualitative behavior of the spring as it depends upon the parameter $.$

Set $m=1,c=0\mathrm{.}3$ and $=1.$ Derive the equation of motion for this oscillator.

Numerical solution. Fix the driving frequency $=1\mathrm{.}25$ and initial condition $x(0)=1,x^{}(0)=0$ and investigate the role of forcing amplitude $.$ Plot the $($a$)$ numerical solution and $($b$)$ phase portrait $(x$ vs$.x^{})$ over a long time period tin$[0,400].$ This allows any transient solutions to sufficiently decay. Describe the solution.

$($a$)=0\mathrm{.}2$

$($b$)=0\mathrm{.}3$

$($c$)=0\mathrm{.}315$

$($d$)=0\mathrm{.}37$

$($e$)=0\mathrm{.}5$

$($f$)=0\mathrm{.}8$

Please Use Matlab!!