Question

For the system below and considering the first quadrant (i.e., the set Q1= {(x,y)∈R2∣x≥0,y≥0}) of the phase plane, answer the following questions. Note that questions (b - e) calls for also negative marking. (a) Sketch the null-clines of the system. (b) Clearly identify the critical points. (c) Indicate the crossings over the null-clines. (d) Show and briefly explain the general orientation of solution curves in the different regions. (e) Identify a subset S of Q1 which is invariant (w.r.t. solution paths of the system of ODE) and which contains the point (5,2) but not the point (1,1). Write your answer in set builder notation. dtdx=x(3−2x);dtdy=y(2y−x2+4)

Answer 1

The nullclines, critical points, crossings, and orientation of solution curves for a given two-dimensional dynamical system have been analyzed. Additionally, an example invariant subset S containing (5,2) but not (1,1) could be S = (x, y) ∈ Q1 .

Step-by-step explanation:

For the given system of equations, the nullclines can be found by setting the derivatives to zero separately. For dx/dt, the nullcline is given by x(3 - 2x) = 0, yielding x = 0 and x = 1.5. For dy/dt, we have y(2y - x^2 + 4) = 0, leading to y = 0 and 2y = x^2 - 4.

Critical points occur where both nullclines intersect. Setting x(3 - 2x) and y(2y - x^2 + 4) to zero simultaneously gives the points (0,0) and (2,2).

Regarding the crossing over the nullclines, the direction of the vector field on either side of the nullcline indicates how the solution curves cross over. This can be determined by evaluating the sign of the derivatives just beside the nullclines.

The orientation of solution curves depends on the signs of the derivatives. If dx/dt > 0, solutions move to the right, and if dy/dt > 0, they move upward. The orientation will change when crossing over nullclines.

An invariant subset S in Q1 containing the point (5,2) but not (1,1) can be defined by the conditions that keep solution paths within the desired area. For instance, a set such as S = (x, y) ∈ Q1 might satisfy these conditions, depending on the vector field.