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2. (a) A dynamical written canonical form gives as SY/ with (:) and k # real constant Identify the type and stability of the critical point at the origin, depending on all possible values giving the ranges values of for which they occur. Justify your ariswer. (b) Consider the dynamical system (=(=) (i) Sketch its phase portrait (ii) Given the linear transformation sketch the phase portrait of the given system give the location< the axes (3/1 that transformed plane. Justify your answer (iii) State one difference and one similarity between the two sketched phase portraits (14 marks) linear system day -2rd) (a) Verify that there eritical point the origin and the Linearisation Theorem to identify its stability properties (b) Use the Poincaré-Bendixson theorem to show that there limit evele in the annulus (There no need find the sense of direction of the limit cycle trajectory.) (15 marks) Two species are competing for common food supply ina given eco-system, with population numbers x1(t) > 0, x2(t) >o at time Their growth can be modeled by the system of almost linear equations, (a) Determine the coordinates ofthe critical points for this system and use the Linearisation Theorern possible, study their type and stability. (b) Obtain the equations of the nullclines of horizontal and vertical crossing directions It a sketch show the nullclines and some trajectories crossing them Henae, using the same sketch draw an approximate phase portrait with allthe infor mation you have from above (no more calculations are noedod). (c) Indicate the point X(0) ( in your diagram and hence comment on what happern to the growth both species for initial populations x1(0) 0.5 x2(0) =0.5. (18 marks) 7. The non-linear dynamical system given by the difforence equation (a) Show that there are two fizued points, x° =2 andr'=b (b) Analyse the stability nature of the fixed point I*=2 for all values of except for b=6. (c) Analyse the stability nature of the fixod point I' = b for all values of except for (d) Using your results from parts (b) and(c) otherwise identify the typec bifurcations that occur and sketch bifurcation dingram If flip bifurcations are identified state if there will be any corresponding two-cycles in this non-linear system and that case predict their stability nature Justify your arawer (12 marks)

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