Fixed point stability for an iteration

A fixed point $x_{\ast }$ of $x_{n+1}=g(x_n)$ is locally asymptotically stable when $|g\prime (x_{\ast })|<1$.

For $\alpha >\beta >0$ the positive fixed point is $(\alpha -\beta {)}^{1/\gamma }$ with multiplier $1-\gamma (1-\beta /\alpha )$. It is stable for $1<\gamma <{\gamma }_{\ast }$ and crosses through minus one at the displayed threshold. Since ${\gamma }_{\ast }>2$, every positive fixed point is stable when $1<\gamma \le 2$. The increasing power controls overcompensatory density dependence at high abundance. A multiplier crossing identifies the local flip mechanism; a separate nonlinear calculation determines its criticality and any resulting attractor.

A hyperbolic sink fixed point of a smooth finite-dimensional map has every Jacobian eigenvalue inside the unit circle. Nearby iterates converge to it, and zero multipliers are allowed for a noninvertible map. Unit-modulus multipliers require nonlinear or exact analysis rather than this strict linear criterion. A complex pair gives spiral contraction, while negative real multipliers alternate directions.

A hyperbolic source fixed point of a smooth finite-dimensional map has every Jacobian eigenvalue outside the unit circle. Forward iteration repels nearby states. For an ordinary differential equation the analogous test instead uses positive real parts, so a source equilibrium criterion should not be substituted into a discrete map calculation.

For every $y_0\ge 0$, this recurrence relation converges to the unique nonnegative fixed point $2$. If $0\le y<2$, then $y<\sqrt{2+y}<2$, since $2+y-y^2=(2-y)(1+y)>0$. If $y>2$, then $2<\sqrt{2+y}<y$. Thus the iterates form a monotone bounded sequence, and continuity forces their limit to be $2$. A cobweb plot alternates vertical steps to $v=\sqrt{2+u}$ and horizontal steps to $v=u$, approaching the same intersection from either side. More quantitatively,so the error keeps its sign and decreases in absolute value by at least the uniform factor $1/(2+\sqrt{2})$. The asymptotic factor is $1/4$.

A planar map has a hyperbolic saddle fixed point when one Jacobian multiplier has modulus less than one and the other has modulus greater than one. In diagonal coordinates the corresponding linear iterates contract and expand geometrically. For a smooth local diffeomorphism, this splitting continues to local stable and unstable manifolds. It differs from the determinant-negative test for a planar continuous flow: an area-preserving map can have a saddle with positive determinant one, because its multipliers can be real reciprocal numbers. The criterion uses the unit circle, not the sign of the real parts.

The fixed points of this cubic iteration of a map satisfy $u=0$ or $u^2=1-1/\lambda$. The nonzero real pair exists for $\lambda <0$ or $\lambda >1$. Its derivative multipliers are $\lambda$ at zero and $3-2\lambda$ at either nonzero point. The fixed point stability for an iteration criterion therefore proves local asymptotic stability of zero for $-1<\lambda <1$ and of the nonzero pair for $1<\lambda <2$. A multiplier of modulus one requires a nonlinear argument rather than this strict derivative test.

A cobweb plot alternates vertical moves to the graph $y=g(x)$ and horizontal moves to the diagonal $y=x$, displaying the iterates of $x_{n+1}=g(x_n)$ geometrically.