Question

Prove that the system of differential equations $$ \dot{x}=x+P(x, y), \quad \dot{y}=y+Q(x, y) $$ where \(P\) and \(Q\) are bounded functions on the entire plane, has at least one equiiibrium position.

Short answer

Question: Prove that the given system of differential equations has at least one equilibrium position. Answer: Using Brouwer's Fixed Point Theorem and the bounds on \(P(x, y)\) and \(Q(x, y)\), we proved that there exists at least one equilibrium position in the square \([-M,M]\times[-N,N]\).

Step-by-step solution

Identify the conditions for an equilibrium position

To find an equilibrium position, we must solve the following system of equations: $$ \begin{cases} x + P(x, y) = 0 \\ y + Q(x, y) = 0 \end{cases} $$

Bound \(P(x, y)\) and \(Q(x, y)\)

Since \(P\) and \(Q\) are bounded functions, there exists constants \(M\) and \(N\) such that \(|P(x, y)| \leq M\) and \(|Q(x, y)| \leq N\) for all \((x, y)\). We can use these bounds to find ranges for \(x\) and \(y\) that ensure an equilibrium position exists.

Finding ranges for \(x\) and \(y\)

To find the possible ranges for \(x\) and \(y\), we can use their respective differential equations: $$ \begin{cases} x + P(x, y) = 0 \implies x = -P(x, y) \\ y + Q(x, y) = 0 \implies y = -Q(x, y) \end{cases} $$ Applying the bounds for \(|P(x, y)| \leq M\) and \(|Q(x, y)| \leq N\), we obtain the following inequalities: $$ \begin{cases} -M \leq x \leq M \\ -N \leq y \leq N \end{cases} $$

Proving the existence of an equilibrium position

We can now prove that an equilibrium position exists using the bounds we found for \(x\) and \(y\). Since \(P(x, y)\) is continuous (as it is bounded) and \(Q(x, y)\) is also continuous, we can use Brouwer's Fixed Point Theorem to guarantee the existence of a fixed point \((x^*, y^*)\) in the compact square \([-M,M]\times[-N,N]\). Then: $$ \begin{cases} x^* + P(x^*, y^*) = 0 \\ y^* + Q(x^*, y^*) = 0 \end{cases} $$ Thus, we have shown that there exists at least one equilibrium position in the given system of differential equations.

Key concepts

Bounded Functions

To begin our journey into the intriguing world of differential equations, we need to understand the concept of bounded functions. A function is said to be bounded if there is a finite limit to its values as it extends over its entire domain. In simple terms, this means that for a bounded function, you can find a real number, say \( M \), such that the absolute value of the function never exceeds \( M \), no matter what input you plug into it.

Think of it like a fence that keeps all the function's values contained within a certain range. This is crucial when solving differential equations because knowing this limit can help us predict how the system behaves and ensure that the solution doesn't lead to any unexpected 'infinity' issues. In our exercise, both \( P(x, y) \) and \( Q(x, y) \) are bounded, which serves as the foundation for proving an equilibrium position exists.

Continuous Functions

Now, let’s explore the smooth universe of continuous functions. A function is continuous when you can draw its graph without lifting your pencil off the paper. More formally, a function \( f \) is continuous at a point \( x_0 \) if the limit of \( f(x) \) as \( x \) approaches \( x_0 \) is equal to \( f(x_0) \).

Why is continuity important for us? Because it guarantees that small changes in the input lead to small changes in the output, which is a desirable property when dealing with differential equations. The solutions tend to behave nicely and predictably within the continuous framework. As mentioned in our exercise, since \( P \) and \( Q \) are bounded on the entire plane, they are also continuous there. This continuity plays a pivotal role in applying Brouwer's Fixed Point Theorem.

Brouwer's Fixed Point Theorem

Brouwer's Fixed Point Theorem is like a magical trick in the mathematician’s toolkit. The theorem states that in any bounded, closed, and convex subset of Euclidean space, any continuous function mapping the set into itself has at least one fixed point. This is a fixed point where the function does not move the input at all; it’s like playing tag with yourself and saying 'Tag! You're it!' without moving an inch.

In the context of our exercise, we use Brouwer's Fixed Point Theorem to prove that within the compact square \([-M,M] \times [-N,N]\) formed by the bounds of the bounded functions, there must be at least one point where the system does not change, i.e., an equilibrium position. This theorem is vital for showing the existence of solutions without actually finding them explicitly.

Ordinary Differential Equations

Finally, let’s tackle ordinary differential equations (ODEs), the centerpiece of our current exploration. ODEs are equations involving functions and their derivatives. They are 'ordinary' because they deal with functions of a single variable, unlike partial differential equations (PDEs) that involve multiple variables.

Why study ODEs? Because they are the bread and butter of modeling the world around us – from the orbits of planets to the dynamics of economies. They help us understand how systems evolve over time based on their current state and changes to that state. The system of differential equations presented in our exercise models a two-dimensional dynamic system where the rate of change of each variable is given by the functions \( \dot{x} \) and \( \dot{y} \). Finding an equilibrium position in such a system is like finding a point of perfect balance, where the forces at play come to a harmonious standstill.