Question

Consider the system of equations $$ d x / d t=y-x f(x, y), \quad d y / d t=-x-y f(x, y) $$ where \(f\) is continuous and has continuous first partial derivatives. Show that if \(f(x, y)>0\) in some neighborhood of the origin, then the origin is an asymptotically stable critical point, and if \(f(x, y)<0\) in some neighborhood of the origin, then the origin is an unstable critical point. Hint: Construct a Liapunov function of the form \(c\left(x^{2}+y^{2}\right) .\)

Short answer

Question: Determine the stability of the origin for the given system of differential equations depending on the sign of the unknown function \(f(x, y)\). Answer: If \(f(x, y) > 0\) in some neighborhood of the origin, the origin is an asymptotically stable critical point. If \(f(x, y) < 0\) in some neighborhood of the origin, the origin is an unstable critical point.

Step-by-step solution

Identifying Given Information and Liapunov Function

We have a system of differential equations $$ \begin{aligned} \frac{dx}{dt} &= y - xf(x, y) \\ \frac{dy}{dt} &= -x - yf(x, y), \end{aligned} $$ where \(f(x, y)\) is continuous and has continuous first partial derivatives. We are asked to determine the stability of the origin \((0, 0)\). For this purpose, we will consider a Liapunov function of the form \(V(x, y) = c(x^2 + y^2)\), where \(c\) is a constant.

Calculate the Time Derivative of the Liapunov Function

To study the stability of the origin, we need to find the time derivative of the Liapunov function. We have $$ \begin{aligned} \frac{dV}{dt} &= \frac{\partial V}{\partial x} \frac{dx}{dt} + \frac{\partial V}{\partial y}\frac{dy}{dt} \\ &= 2cx \left(y - xf(x, y)\right) + 2cy\left(-x - yf(x, y)\right). \end{aligned} $$

Analyze the Time Derivative of the Liapunov Function

Now, we need to analyze the sign of \(\frac{dV}{dt}\) to determine the stability of the origin. Let's consider the two cases: Case 1: \(f(x, y) > 0\) in some neighborhood of the origin. In this case, we have $$ \begin{aligned} \frac{dV}{dt} &= 2cx(y - x f(x, y)) - 2cy(x + y f(x, y))\\ &= -2cxyf(x, y) < 0. \end{aligned} $$ Hence, the origin is an asymptotically stable critical point in this case. Case 2: \(f(x, y) < 0\) in some neighborhood of the origin. In this case, we have $$ \begin{aligned} \frac{dV}{dt} &= 2cx(y - x f(x, y)) - 2cy(x + y f(x, y))\\ &= -2cxyf(x, y) > 0. \end{aligned} $$ Hence, the origin is an unstable critical point in this case.

Conclusion

We have analyzed the system of equations given, and shown that by constructing a Liapunov function of the form \(c(x^2 + y^2)\), we can conclude: if \(f(x, y) > 0\) in some neighborhood of the origin, the origin is an asymptotically stable critical point; whereas if \(f(x, y) < 0\) in some neighborhood of the origin, the origin is an unstable critical point.

Key concepts

Liapunov Function

In the context of differential equations, particularly when analyzing the stability of critical points, a Liapunov function is a powerful tool. Created by the Russian mathematician Aleksandr Lyapunov, it serves as a mathematical 'energy-like' measure for systems described by differential equations.

Think of a Liapunov function as a sort of 'gauge' that can measure how the system's state changes over time. If we find a suitable Liapunov function, we can tell whether a critical point, such as the equilibrium of a system, is stable or not. The function is typically a scalar, and for a two-dimensional system, a common choice is a quadratic form like V(x, y) = c(x^2 + y^2), which represents a paraboloid in a 3D space.

The clever trick here is to choose a Liapunov function that, when we do the math, will have a time derivative that reveals something about the stability. If this derivative is negative in the vicinity of a critical point, we're holding a sign that points to stability, hinting that the system wants to stay put at equilibrium.

Asymptotically Stable Critical Point

Now, what exactly do we mean when we say a critical point is asymptotically stable? Imagine a ball settling at the bottom of a bowl. No matter how you nudge it, it always finds its way back to the center. An asymptotically stable critical point behaves similarly in the realm of differential equations.

For a critical point to be asymptotically stable, two criteria must be met: first, the system must return to the critical point if it's given a slight nudge (we call this stability), and second, the system's state must approach the critical point as time goes on (this is the 'asymptotic' part).

Back to our exercise, if the function f(x, y) is positive near the origin, any deviations will dwindle away over time, and the state will gravitate back to the origin, satisfying both criteria for asymptotic stability.

Time Derivative

The time derivative is essentially the rate at which a function changes with respect to time. When analyzing the system's stability with a Liapunov function, what matters is how the 'energy' (or the Liapunov function's value) changes as the system evolves.

For instance, if the system's energy decreases over time (meaning the time derivative of the Liapunov function is negative), then the system is losing its 'oomph' to stray far from the critical point - a hint that the point is stable. Conversely, if that energy tends to increase (positive time derivative), we're looking at a system that's eager to deviate from the critical point, spelling instability.

Mathematically, the time derivative of our chosen Liapunov function V(x, y) = c(x^2 + y^2) in our system's context was found by applying the chain rule, summing the partial derivatives, and setting the stage for assessing stability.

System of Differential Equations

A system of differential equations like the one we're dealing with can be seen as a set of equations that describes the behavior of a multi-dimensional system. Each equation represents the rate of change of one aspect of the system, typically with respect to time.

These systems can model a vast array of phenomena, from the orbits of planets to the way populations of animals change through time. The key to understanding them lies in the interplay between their individual components - in our case, how the variable x and y influence each other.

Any equilibrium points found within these systems, where the rates of change are zero, become critical to understanding the overall system behavior. Stability analysis using Liapunov functions is just one of the many ways we can peer into the system's future behavior without solving the equations explicitly.