see attachment

Non Linear Systems Stability
For each of the questions below, we classify using linearization method into asymptotic stability, neutral stability &
unstable systems. Initially, linearize the system xi = Fi(xi, yi) by dxi =
∂Fi
∂xi
to get Jacobian matrix J. Neglecting
O(x2), we can write about stationary points that
[
x′1
x′2
]
=J
[
x1
x2
]
Now, a system is stable about staionary point if
Eigen Values of the Jacobian about it is ≤ 0, and unstable if else. Three behaviours of Linear systems about critical
points are as follows.
• All solutions converge to the origin/critical point as t→∞. This happens when the Eigen- values are negative
or the have negative real part. In this case we call the origin asymptotically stable.
• Solutions near the origin/critical point stay near the origin for all time & are bounded. This happens when
the eigenvalues are purely complex or there is an eigenvalue which is zero while the other is negative. In this
case we call the origin stable.
• If neither of the above two occur, we call the origin unstable. That is, at least one trajectory leaves the
vicinity of the origin.
So, for this calculate β = Tr A, γ = det A & δ = β2 − 4γfor the corresponding Jacobians & we know Eigen values
λ’s are solution of λ2 − λβ + γ = 0 which is 12 (λ ±
√
β2 − 4γ). Then, sketch approximate phase diagrams in the
vicinity of each stationary state.
Then evaluate Eigen system to make the picture more precise(and do this iff Eigen Values are real). Finally,
reconstruct the global phase diagram.
1. Given
x′1 = F1(x, y) = −x31 + x2 (1)
x′2 = F2(x, y) = x1 − x2 (2)
• First we get...