Dynamical Systems in Neuroscience:

88 One-Dimensional SystemsVF (V)1VVF (V) 2F (V)1VF (V) 2VF (V) 2F (V)1VabcF (V)1VF (V)1VF (V)1VF (V) 2VF (V) 2VF (V) 2VdefFigure 3.37: Which of the pairs correspond to topologically equivalent dynamical systems?(All intersections with the V axis are marked as dots.)5. Draw phase portraits of the systems in Fig. 3.37. Which of the pairs in the figurecorrespond to topologically equivalent dynamical systems?6. (Saddle-node bifurcation) Draw the bifurcation diagram and representative phaseportraits of the system ẋ = a + x 2 , where a is a bifurcation parameter. Find theeigenvalues at each equilibrium.7. (Saddle-node bifurcation) Use definition in Sect. 3.3.4 to find saddle-node bifurcationpoints in the following systems:(a) ẋ = a + 2x + x 2 ,(b) ẋ = a + x + x 2 ,(c) ẋ = a − x + x 2 ,(d) ẋ = a − x + x 3 (Hint: verify the non-hyperbolicity condition first),(e) ẋ = 1 + ax + x 2 ,(f) ẋ = 1 + 2x + ax 2 ,where a is the bifurcation parameter.8. (Pitchfork bifurcation) Draw the bifurcation diagram and representative phaseportraits of the system ẋ = bx − x 3 , where b is a bifurcation parameter. Find theeigenvalues at each equilibrium.