We may use Lagrange points to find the general stable and
unstable equilibrium points of Cournot model in the incomplete market.
However, this delay in the model can stabilize an initially
unstable equilibrium of the system (see Theorem 12, (3)(b)(ii)), which represents a major economic interest since it allows the saving to get rid of risks from the cyclic growth.
(i)If g([a.sup.2]M + [b.sup.2] g) < 0, then [e.sup.M.sub.1] is an
unstable equilibrium point.
For [R.sup.*.sub.thr] < [R.sup.*.sub.1] < 1, system (1) will present the locally asymptotically stable low-criminality equilibrium [P.sub.l] plus two positive high-criminality equilibria, [P.sup.+.sub.h] and [P.sup.-.sub.h], which will correspond to the solutions of (9): [C.sup.+.sub.2], the higher solution, which corresponds to the stable equilibrium, and [C.sup.-.sub.2], the smaller solution, which corresponds to the
unstable equilibrium.
So the drop is in a state of
unstable equilibrium, which can be easily disturbed by breaking the tail.
Moreover, for each b < [b.sub.0], but close to [b.sub.0], there is a stable limit cycle close to the
unstable equilibrium point [E.sub.0].
It can be found that this new chaotic system has two isolated chaotic attractors or two disconnected chaotic attractors (named "positive attractor" and "negative attractor" in this paper), which depend on the distance between the initial points (initial conditions) and the
unstable equilibrium points.
If an attractor's domain of attraction does not intersect with any
unstable equilibrium, it is called a hidden attractor; otherwise it is called a self-excited attractor [14-16].
When [phi] < 0 with [absolute value of ([phi])] [much less than] 1, 0 is locally asymptotically stable, and there exists a positive
unstable equilibrium; when 0 < [phi] [much less than] 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.
Bourguignon (1981) shows that, for a given proportion of rich [a.sub.2] satisfying 0 < [[alpha].sub.2] < A, the possible equilibria are pairs ([k.sub.1], [k.sub.2]) each consisting of an
unstable equilibrium [k.sub.1] and a stable equilibrium [k.sub.2] with [k.sub.2] > [k.sub.1].
This suggests the possibility of an
unstable equilibrium that is analogous to the "tipping point" at x' in the standard model.