Conservative Dynamical Systems of 1 DoF
Example : Eardrum oscillations  x'' = -x - a x^2
A. Solutions

System Definition, Potential and energy Integral

f[x_]=-x-a x^2
V=-∫f[x]\[DifferentialD]x
Energy[x_,y_]=(1/2)y^2+V

Plot[V/.{a->1/4}, {x,-8,3}]

Equilibrium Points

eqp=Solve[f[x]==0,x]

ddV=D[V,{x,2}]
ddV/.eqp[[1]]
ddV/.eqp[[2]]

Analytic Solution

deq1=x'[t]==y[t]
deq2=y'[t]==f[x[t]]

DSolve[{deq1,deq2},{x,y},t]

DSolve[{deq1,deq2,x[0]==x0, y[0]==y0},{x,y},t]

Numerical Solution

deq2a=deq2/.{a->1/4}
x0=1.5;
y0=0.1;
tmax=20;
nsol=NDSolve[{deq1,deq2a,x[0]==x0, y[0]==y0},{x,y},{t,0,tmax}]

xt=x[t]/.nsol[[1]]
yt=y[t]/.nsol[[1]]
Plot[{xt,yt},{t,0,tmax}, PlotStyle->{Blue,Gray}]