TABLE OF CONTENTS | 4.2.1 | Analyze the case  where f( x) = has no real zeros. |
| 4.2.2 | Draw all the possible one-dimensional phase-space solution behaviors for three distinct fixed-points. |
| 4.3.1 | The logistic differential equation, as given by Eq. (4.3.7), can be solved exactly. Find this solution and use it to check on the correctness of the conclusions reached in section 4.3.2 In particular, discuss the nature of the solution if x(0) < 0. Plot x( t) vs t for x(0) < 0. |
| 4.3.2 | The differential equation, given by Eq. (4.3.12), can be solved exactly. To show this, multiply both sides by 2 x and use the new variable z = x 2. Carry out this calculation and verify that the discussion presented in section 4.3.3 is correct. |
| 4.3.3 | Plot the fixed-points and typical trajectories for the example in section 4.3.4 for ?5 ? x ? 5. |
| 4.3.4 | Analyze the phase-space and the trajectory ( x vs t) space for the differential equation  |
| 4.3.5 | Extend the method given in section 4.3.5 to find the general qualitative properties of the solutions to the differential equations [1]; [2]  |
| 4.4.1 | The fixed-point, ( x,y) = (0, 0), for the system of equations  is a saddle point. Show that a small linear perturbation to this equation leaves the nature of the fixed-point unchanged. |
| 4.4.2 | The system of equations  has a... |
