Final answer:
To find the general solution for the given system, we can find the eigenvalues and eigenvectors of the coefficient matrix. The equilibrium solution 0 is a sink and stable. The v-nullcline is -x1 + x2 = 0 and the h-nullcline is -2x1 - x2 = 0. The phase portrait can be sketched using the nullclines as an aid.
Step-by-step explanation:
To find the general solution for the system x′ = [−1 1 −2 −1]x, we can first find the eigenvalues and eigenvectors of the coefficient matrix. The characteristic equation is det(A - λI) = 0, where A is the coefficient matrix, λ is the eigenvalue, and I is the identity matrix. Solving this equation, we find λ1 = -1 and λ2 = -1. The corresponding eigenvectors are v1 = [1, -2] and v2 = [1, -1]. The general solution is x = c1*e^(-t)*[1, -2] + c2*e^(-t)*[1, -1], where c1 and c2 are arbitrary constants.
The equilibrium solution 0 is a sink because both eigenvalues are negative, indicating that the system will approach 0 as time goes on. The equilibrium solution 0 is stable because small perturbations around 0 will decay over time.
The v-nullcline is the set of points where the first component of x' is equal to 0. In this system, the v-nullcline is represented by the equation -x1 + x2 = 0. The h-nullcline is the set of points where the second component of x' is equal to 0. In this system, the h-nullcline is represented by the equation -2x1 - x2 = 0.
The phase portrait can be sketched using the nullclines as an aid. Plot the v-nullcline, which is a straight line passing through the origin with a slope of 1. Plot the h-nullcline, which is a straight line passing through the origin with a slope of -2. The intersection of the nullclines represents the equilibrium point at 0, and the direction of the arrows on the nullclines indicates the movement of the system.