Question

We will prove part of Theorem 9.3 .2 : If the critical point \((0,0)\) of the almost linear system $$ d x / d t=a_{11} x+a_{12} y+F_{1}(x, y), \quad d y / d t=a_{21} x+a_{22} y+G_{1}(x, y) $$ is an asymptotically stable critical point of the corresponding linear system $$ d x / d t=a_{11} x+a_{12} y, \quad d y / d t=a_{21} x+a_{22} y $$ then it is an asymptotically stable critical point of the almost linear system (i). Problem 12 deals with the corresponding result for instability. In this problem we show that the Liapunov function constructed in the preceding problem is also a Liapunov function for the almost linear system (i). We must show that there is some region containing the origin for which \(\hat{V}\) is negative definite. (a) Show that $$ \hat{V}(x, y)=-\left(x^{2}+y^{2}\right)+(2 A x+B y) F_{1}(x, y)+(B x+2 C y) G_{1}(x, y) $$ (b) Recall that \(F_{1}(x, y) / r \rightarrow 0\) and \(G_{1}(x, y) / r \rightarrow 0\) as \(r=\left(x^{2}+y^{2}\right)^{1 / 2} \rightarrow 0 .\) This means that given any \(\epsilon>0\) there exists a circle \(r=R\) about the origin such that for \(0

Short answer

In conclusion, we have shown that if the origin is asymptotically stable for the linear system, then it is asymptotically stable for the almost-linear system, as well. We have derived the Liapunov function, \(\hat{V}(x, y)\), and confirmed that \(\dot{V}(x, y) < 0\) for a sufficiently small \(r\), ultimately proving the origin's asymptotic stability in the given almost-linear system.

Step-by-step solution

Find the expression for \(\hat{V}(x, y)\)

We are given the following expression for \(\hat{V}(x, y)\): $$ \hat{V}(x, y)=-\left(x^{2}+y^{2}\right)+(2 A x+B y) F_{1}(x, y)+(B x+2 C y) G_{1}(x, y) $$ where \(F_1(x, y)\) and \(G_1(x, y)\) are the nonlinear terms of the almost-linear system.

Prove that \(\dot{V}(x, y) < 0\) for sufficiently small \(r\)

To find the expression for \(\dot{V}(x, y)\), we first need to express \(x\) and \(y\) in polar coordinates: $$ x = r\cos\theta $$ $$ y = r\sin\theta $$ Now, recall the conditions given in the problem: $$ \lim\limits_{r\to 0} \frac{F_{1}(x, y)}{r} = 0 $$ $$ \lim\limits_{r\to 0} \frac{G_{1}(x, y)}{r} = 0 $$ Given any \(\epsilon > 0\), we need to find \(R\) such that for \(0<r<R, \left|F_{1}(x, y)\right|<\epsilon r\) and \(\left|G_{1}(x, y)\right|<\epsilon r\). Let \(M = \max\{|2A|,|B|,|2C|\}\), and choose \(\epsilon\) small enough such that: $$ \epsilon < \frac{1}{M} $$ Then, for \(0<r<R\), we have: $$ |(2Ax+By)F_1(x,y)| < M\epsilon r^2 < r^2 $$ $$ |(Bx+2Cy)G_1(x,y)| < M\epsilon r^2 < r^2 $$ Thus, $$ \dot{V}(x, y) = -(x^2+y^2) + (2Ax+By)F_1(x,y) + (Bx+2Cy)G_1(x,y) < 0$$ for \(0<r<R\). This proves that the critical point \((0, 0)\) is asymptotically stable for the almost-linear system.

Key concepts

Liapunov Function

A Liapunov function is a mathematical tool used to demonstrate the stability of equilibrium points within dynamical systems. In essence, it can be thought of as an energy-like function that decreases over time, indicating that the system's state is approaching a stable point. For a dynamical system given by
\[\begin{equation}\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}),\end{equation}\]
a candidate Liapunov function, \(V(\mathbf{x})\), is chosen such that \(V(\mathbf{x}) > 0\) for all \(\mathbf{x} eq \mathbf{0}\) and \(V(\mathbf{0}) = 0\). The rate of change of \(V\) along the system trajectories, denoted as \(\dot{V}(\mathbf{x})\), must satisfy \(\dot{V}(\mathbf{x}) < 0\) for all \({\mathbf{x} eq \mathbf{0}\) for the critical point to be asymptotically stable. In the exercise, you are tasked with showing that a proposed \(\hat{V}(x, y)\) function can serve as a Liapunov function for the almost linear system by proving that its derivative is negative definite in the vicinity of the origin.

Almost Linear Systems

An almost linear system refers to a nonlinear dynamical system that can be approximated by a linear one in the neighborhood of an equilibrium point. The system is characterized by equations of the form
\[\begin{equation}\dot{x}=a_{11}x+a_{12}y+F_{1}(x, y), \dot{y}=a_{21}x+a_{22}y+G_{1}(x, y)\end{equation}\]
where \(F_1\) and \(G_1\) represent the nonlinear parts of the system. The study of almost linear systems often starts with investigating the stability of the corresponding linear system. If the linear part has a stable critical point, then under certain conditions, the same can be true for the nonlinear system. One of those conditions can be established using a Liapunov function, as shown in the provided exercise, where one shows that \(\hat{V}(x, y)\) effectively captures the system's behavior near the critical point.

Polar Coordinates

The use of polar coordinates is a powerful technique particularly when dealing with two-dimensional systems involving rotations or radially symmetric components. In polar coordinates, a point \((x, y)\) in the plane is represented by a radius \(r\) and angle \(\theta\), where
\[\begin{equation}x = r \cos(\theta), \y = r \sin(\theta).\end{equation}\]
In dynamics, this transformation is beneficial for simplifying the analysis of circular motion or when considering systems that are easier to express in terms of distance from a central point and angular displacement. This approach is used in the problem to analyze the behavior of the almost linear system near the origin, allowing for a clearer perspective on how the perturbations \(F_1\) and \(G_1\) diminish in relation to the radius \(r\), assisting in proving the stability via the Liapunov function.

Nonlinear Dynamical Systems

Nonlinear dynamical systems are characterized by equations where the rate of change of the system's state is not a linear function of the state itself. This nonlinearity can lead to complex behaviors such as chaos, multiple equilibria, and sensitivity to initial conditions not observed in linear systems. Studying the stability of critical points in nonlinear systems is more involved due to these potential complexities. Thus, the identification of a Liapunov function, as discussed in the exercise, becomes an important route to establishing stability conditions without solving the system explicitly. In practice, the stability of these systems can have critical implications across various scientific and engineering disciplines, including control theory, economics, ecological modeling, and beyond.