Are Saddle Points Stable

Daniel Brown
• Sunday, 03 January, 2021
• 7 min read

An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle.

saddle point unstable stable equation its manifolds
(Source: www.researchgate.net)


For example, the point (0,0,0){\display style (0,0,0)} is a saddle point for the function z=x4y4,{\display style z=X{4}by{4}, } but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian.

Media related to Saddle point at Wikimedia Commons But saddle -path stable systems have found important uses in economics, because this feature of theirs accommodates purposeful behavior.

Assume that an economic system described by a set of difference equations was properly stable in the full mathematical sense. That would imply that no matter where we started, automatically the system would tend to its fixed point/long-run equilibrium.

But if the system is saddle -path stable “, then agents track the trajectory followed, and if it appears to deviate from the one that leads to the fixed point, they purposefully change their own behavior in order to go back to the desired trajectory, since corner solutions are not optimal. For hyperbolic equilibrium of autonomous vector fields, the linearization captures the local behavior near the equilibrium for the nonlinear vector field.

saddle node unstable stable fixed state curve ing points down diagram
(Source: www.researchgate.net)

Suppose \((x_{0}, y_{0})\) is a hyperbolic equilibrium point of this vector field, i.e. the two eigenvalues of the Jacobean matrix: More precisely, there exists a neighborhood U of \((x_{0}, y_{0})\) such that for any \(p \in U\), \(\phi_{t}(p)\) leaves U as t increases.

More precisely, there exists a neighborhood S of \((x_{0}, y_{0})\) such that for any \(p \in S\), \(\phi_{t}(p)\) approaches \((x_{0}, y_{0})\) at an exponential rate as t increases. In particular, we will need to transform (6.1) to a coordinate system that “localizes” the behavior near the equilibrium point and specifically displays the structure of the linear part.

Reflects the hyperbolic nature of the equilibrium point. We now state how this saddle point structure is inherited by the nonlinear system by stating the results of the stable and unstable manifold theorem for hyperbolic equilibrium for two-dimensional nonautonomous vector fields.

It is locally invariant in the sense that any trajectory starting on the curve approaches the origin at an exponential rate as \(t \right arrow ifty\), and it leaves \(B_{\epsilon}\) as \(t \right arrow - ifty\). Moreover, the curve satisfying these three properties is unique.

From this calculation we can conclude that the origin is a hyperbolic saddle point. Moreover, the x-axis is the unstable subspace for the linearized vector field and the y-axis is the stable subspace for the linearized vector field.

stable node unstable focus saddle point
(Source: www.researchgate.net)

It is easy to compute the flow generated by (6.16). The x-component can be solved and substituted into the y component to yield a first order linear nonautonomous equation.

We now compute the global invariant manifold structure for each of the equilibrium, beginning with (0, 0). The x-axis is clearly the global stable manifold for this equilibrium point.

In this example we consider the following nonlinear autonomous vector field on the plane: We want to classify the linearized stability of the equilibrium.

We evaluate this expression for the eigenvalues at each of the equilibrium to determine their linearized stability. Therefore the eigenvalues are always real and of opposite sign.

This implies that these two fixed points are always sinks. \(\delta^2-8 < 0\): The eigenvalues have a nonzero imaginary part.

stable node saddle point case dynamics focus
(Source: www.researchgate.net)

6.4 we sketch the local invariant manifold structure for these two cases. Figure 6.4: Local invariant manifold structure of (6.36).

6.5 we sketch the global invariant manifold structure for the two cases. In the coming lectures we will learn how we can justify this figure.

However, note the role that the stable manifold of the saddle plays in defining the basins of attractions of the two sinks. Figure 6.5: A sketch of the global invariant manifold structure of (6.36).

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