Are Saddle Point Unstable

Carole Stephens
• Saturday, 14 November, 2020
• 11 min read

Saddle point between two hills (the intersection of the figure-eight z{\display style z}-contour)In mathematics, a saddle point or minimal point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extreme of the function. 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.

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


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. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.

In a two-player zero-sum game defined on a continuous space, the equilibrium point is a saddle point. 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. The term saddle -node bifurcation' is most often used in reference to continuous dynamical systems.

saddle unstable point equilibrium example examples systems another
(Source: systems-sciences.uni-graz.at)

Another name is blue sky bifurcation in reference to the sudden creation of two fixed points. If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node).

A typical example of a differential equation with a saddle -node bifurcation is: In fact, this is a normal form of a saddle -node bifurcation.

As can be seen by the animation obtained by plotting phase portraits by varying the parameter {\display style \alpha}, A saddle -node bifurcation also occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from PX{\display style PX} to p{\display style p}, that is, the consumption rate is constant and not in proportion to resource x{\display style x}.

Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. A non-autonomous version of the saddle -node bifurcation (i.e. the parameter is time-dependent) has also been studied.

^ Chong, KET Hing; Samarasinghe, Sandy; Pulaski, Don; Zheng, Die (2015). Computational techniques in mathematical modelling of biological switches.

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

“Time-dependent saddle –node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions”. Elements of Applied Bifurcation Theory (Second ed.).

Computational Techniques in Mathematical Modelling of Biological Switches. (eds) MODSIM2015, 21st International Congress on Modelling and Simulation (MOD SIM 2015).

Modelling and Simulation Society of Australia and New Zealand, December 2015, pp. Kohl, Knot Singh; Ha slam, Michael C. (2018).

Einstein Field Equations as a Fold Bifurcation. Journal of Geometry and Physics Volume 123, January 2018, Pages 434-437.

Eigenvalues and eigenvectors are very useful in the modeling of chemical processes. Differential equations are used in these programs to operate the controls based on variables in the system.

stable saddle point manifold figure solved produces analytical curve shown result shape same
(Source: www.chegg.com)

These equations can either be solved by hand or by using a computer program. The solutions for these differential equations will determine the stability of the system.

For the other two cases, the system will not be able to return to steady state. For the unnamed situation, the constant fluctuation will be hard on the system and can lead to equipment failure.

The final situation, with the ever-increasing amplitude of the fluctuations will lead to a catastrophic failure. There are a couple ways to develop the differential equation used to determine stability.

A second method would be using actual data found from running the system. For the South stability test, calculating the eigenvalues is unnecessary which is a benefit since sometimes that is difficult.

Finally, the advantages and disadvantages of using eigenvalues to evaluate a system's stability will be discussed. Eigenvalues and eigenvectors can be used as a method for solving linear systems of ordinary differential equations (Odes).

saddle point unstable source phase spiral example examples plane another
(Source: systems-sciences.uni-graz.at)

When trying to solve large systems of Odes however, it is usually best to use some sort of mathematical computer program. Mathematical is a program that can be used to solve systems of ordinary differential equations when doing them by hand is simply too tedious.

Once one overcomes the syntax of Mathematical, solving enormous systems of ordinary linear differential equations becomes a piece of cake! A linear system will be solved by hand and using Eigenvalues expression in Mathematical simultaneously.

A stable fixed point is such that a system can be initially disturbed around its fixed point yet eventually return to its original location and remain there. To illustrate this concept, imagine a round ball in between two hills.

If left alone, the ball will not move, and thus its position is considered a fixed point. If left undisturbed, the ball will still remain at the peak, so this is also considered a fixed point.

However, a disturbance in any direction will cause the ball to roll away from the top of the hill. The top of the hill is considered an unstable fixed point.

saddle unstable point stable manifolds dashed solid speed under publication
(Source: www.researchgate.net)

The particular stability behavior depends upon the existence of real and imaginary components of the eigenvalues, along with the signs of the real components and the distinctness of their values. When eigenvalues are of the form, where and are real scalars and is the imaginary number, there are three important cases.

These three cases are when the real part is positive, negative, and zero. In all cases, when the complex part of an eigenvalue is non-zero, the system will be oscillatory.

Positive Real Part When the real part is positive, the system is unstable and behaves as an unstable oscillator. This can be visualized as a vector tracing a spiral away from the fixed point.

The plot of response with time of this situation would look sinusoidal with ever-increasing amplitude, as shown below. This situation is usually undesirable when attempting to control a process or unit.

This can be visualized in two dimensions as a vector tracing a circle around a point. Even so, this is usually undesirable and is considered an unstable process since the system will not go back to steady state following a disturbance.

phase plane analysis nonlinear saddle point stability unstable node
(Source: www.slideshare.net)

This can be visualized as a vector tracing a spiral toward the fixed point. The plot of response with time of this situation would look sinusoidal with ever-decreasing amplitude, as shown below.

This situation is what is generally desired when attempting to control a process or unit. The oscillation will quickly bring the system back to the set point, but will over shoot, so if overshooting is a large concern, increased damping would be needed.

While discussing complex eigenvalues with negative real parts, it is important to point out that having all negative real parts of eigenvalues is a necessary and sufficient condition of a stable system. Complex Part of Eigenvalues As previously noted, the stability of oscillating systems (i.e. systems with complex eigenvalues) can be determined entirely by examination of the real part.

Although the sign of the complex part of the eigenvalue may cause a phase shift of the oscillation, the stability is unaffected. This is just a trivial case of the complex eigenvalue that has a zero part.

On a gradient field, a spot on the field with multiple vectors circularly surrounding and pointing out of the same spot (a node) signifies all positive eigenvalues. Graphically, real and positive eigenvalues will show a typical exponential plot when graphed against time.

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

Graphically, real and negative eigenvalues will output an inverse exponential plot. It is called a saddle point because in 3 dimensional surface plots the function looks like a saddle.

In general, the determination of the system's behavior requires further analysis. If the two repeated eigenvalues are positive, then the fixed point is an unstable source.

If the two repeated eigenvalues are negative, then the fixed point is a stable sink. Below is a table summarizing the visual representations of stability that the eigenvalues represent.

Note that the graphs from Peter Woolf's lecture from Fall'08 titled Dynamic Systems Analysis II: Evaluation Stability, Eigenvalues were used in this table. The process of finding eigenvalues for a system of linear equations can become rather tedious at times and to remedy this, a British mathematician named Edward South came up with a handy little short-cut.

If so, there is at least one value with a positive or zero real part which refers to an unstable node. The way to test exactly how many roots will have positive or zero real parts is by performing the complete South array.

control system modern unstable stable
(Source: www.slideshare.net)

For all the roots of the polynomial to be stable, all the values in the first column of the South array must be positive. If any of the values in the first column are negative, then the number of roots with a positive real part equals the number of sign changes in the first column.

The following image can work as a quick reference to remind yourself of what vector field will result depending on the eigenvalue calculated. The table below gives a complete overview of the stability corresponding to each type of eigenvalue.

There are several advantages of using eigenvalues to establish the stability of a process compared to trying to simulate the system and observe the results. However, there are situations where eigenvalue stability can break down for some models.

What would the following set of eigenvalues predict for the system's behavior? Chemical Process Control: A Time Domain Approach.

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