Consider the unforced mass$-$spring system

$m{y}^{}+g(y)=0$

with three different models for the spring force

hardening spring: $g(y)=k(1+y^2)y$;

softening spring: $g(y)=k(1-y^2)y$;

linear spring: $g(y)=ky,$

and $k>0.$

$($a$)$ Determine a state$-$space representation of this system.

$($b$)$ Find equilibrium points of the above systems. Discuss your observations for three different spring

force models.

$($c$)$ Is this system

causal,

time$-$varying,

linear,

memoryless,

finite$-$dimensional?

Explain.

$($d$)$ For three different spring force models with $m=k=1,$ use Matlab to simulate systems' responses

from four different initial conditions. Choose these four initial conditions in order to demonstrate

different behavior of the models. Plot corresponding results in the phase plane $($horizontal axis de$-$

termined by position $y(t),$ vertical axis determined by velocity $y^{}(t)$ and discuss your observations.

You should plot four different figures and each of these should contain three phase portraits.