(Source: www.researchgate.net)

Contents

- 1. Accommodates
- 2. \\phi_{t}p\
- 3. Inspection
- 4. Equilibrium
- 5. Invariant
- 6. Ever-increasing
- 7. Simultaneously
- 8. Undisturbed
- 9. Oscillator
- 10. Overshooting
- 11. Determination
- 12. Polynomial

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.

(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.

(Source: www.researchgate.net)

By inspection, we see that the y-axis (i.e. x = 0) is the global stable manifold for the origin. It is also tangent to the unstable subspace at the origin.

The x-component can be solved and substituted into the y component to yield a first order linear nonautonomous equation. The global unstable manifold of the origin is the set of initial conditions having the property that the trajectories through these initial conditions approach the origin at an exponential rate as \(t \right arrow - ifty\).

Note that the x and y components evolve independently. We now compute the global stable and unstable manifolds of these equilibrium.

In this example we consider the following nonlinear autonomous vector field on the plane: Note that the x and y components evolve independently.

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.

(Source: www.researchgate.net)

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. This implies that these two fixed points are always sinks.

\(\delta^2-8 < 0\): The eigenvalues have a nonzero imaginary part. 6.4 we sketch the local invariant manifold structure for these two cases.

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).

(Source: www.slideshare.net)

Differential equations are used in these programs to operate the controls based on variables in the system. 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).

(Source: www.slideshare.net)

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.

Note that, in the Mathematical inputs below, “In:=” is not literally typed into the program, only what is after it. The term is used here to more accurately demonstrate coding in Mathematical.

In:= Eigenvalues[ParseError: EOF expected(click for details) As mentioned earlier, we have a degree of freedom to choose for either x or y. Let’s assume that x=1.

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.

(Source: www.researchgate.net)

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.

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.

(Source: www.chegg.com)

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.

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.

(Source: systems-sciences.uni-graz.at)

Although the sign of the complex part of the eigenvalue may cause a phase shift of the oscillation, the stability is unaffected. 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. Graphically, real and negative eigenvalues will output an inverse exponential plot.

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.

(Source: www.slideshare.net)

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.

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. General method that can be applied to a variety of processes.

First, let us rewrite the system of differentials in matrix form. All solutions that do not start at (0,0) will travel away from this unstable saddle point.

(Source: math.stackexchange.com)