The following two$-$dimensional linear system models the dynamics of Romeo and Juliet, where the variables $R$ and $J$ each represent the "strength of attraction" towards the other:

$R^{}=aR+bJ$

$J^{}=bR+aJ$

Let $a<mn\; id="EditorElement\_2369276">0</mn>$ and $b>0,$ such that both Romeo and Juliet are "cautious lovers".

$($a$)$ List at least two modelling assumptions that have been made in the development of this model.

$($b$)$ Without doing any calculations, describe the possible solutions as $t$ in a twodimensional linear system. How many fixed points could a two$-$dimensional linear system have?

$($c$)$ Classify all fixed points of the above model and draw phase portraits of the possible dynamics. Show all working out.

$($d$)$ Interpret the results of part $($c$)$ in terms of Romeo and Juliet's feelings. Does the development of their feelings depend on the initial conditions?

Total: $[20]$

$2.$ For each of the following systems do the following:

$($i$)$ find all the fixed points;

$($ii$)$ determine the stability type $($e$.$g$.$ stable node, unstable focus$)$ of all fixed points as a function of the parameter $$;

$($iii$)$ identify for which parameter values of $$ a bifurcation involving fixed points occurs and identify the bifurcation;

$($iv$)$ sketch the corresponding bifurcation diagram $($with no need to consider potential limit cycles$).$

$($a$)$

$x^{}=x+\frac{x}{1+x^2}$

$($b$)$

$x^{}=y-2x$

$y^{}=+x^2-y$