Question

Consider the system $$ \begin{array}{l} \dot{x}=y-x f(x, y), \\ \dot{y}=-x-y f(x, y), \end{array} $$ where \(f(x, y)\) is analytic at the origin and \(f(0,0)=0\). Describe the relation between \(f(x, y)\) and the type of stability.

Short answer

Answer: The stability of the given system of differential equations at the origin is independent of the behavior of the function f(x, y). The stability is determined by the eigenvalues of the Jacobian matrix, which in this case are purely imaginary, making the fixed point (0,0) semi-stable regardless of the actual behavior of f(x, y).

Step-by-step solution

Find the linearization of the given system

To do this, calculate the Jacobian matrix of the given system of differential equations. The Jacobian matrix \(J(x, y)\) is a 2x2 matrix with elements given by the first partial derivatives of the given functions with respect to x and y: $$ J(x, y) = \begin{pmatrix} \frac{\partial (\dot{x})}{\partial x} & \frac{\partial (\dot{x})}{\partial y} \\ \frac{\partial (\dot{y})}{\partial x} & \frac{\partial (\dot{y})}{\partial y} \end{pmatrix} $$ For the given system: $$ J(x, y) = \begin{pmatrix} \frac{\partial (y-x f(x, y))}{\partial x} & \frac{\partial (y-x f(x, y))}{\partial y} \\ \frac{\partial (-x-y f(x, y))}{\partial x} & \frac{\partial (-x-y f(x, y))}{\partial y} \end{pmatrix} = \begin{pmatrix} -f(x, y)-xf_x(x,y) & 1-xf_y(x,y) \\ -1-yf_x(x,y) & -f(x, y)-yf_y(x,y) \end{pmatrix} $$ Evaluate the Jacobian matrix, \(J(x, y)\) at the origin (0,0): $$ J(0, 0) = \begin{pmatrix} -0 & 1 \\ -1 & -0 \end{pmatrix} $$

Find eigenvalues of the Jacobian matrix at the origin

Calculate the eigenvalues of the Jacobian matrix at the origin by solving for the roots of the characteristic polynomial: $$ \text{det}(J(0, 0) - \lambda I) = \begin{vmatrix} -\lambda & 1 \\ -1 & -\lambda \end{vmatrix} = \lambda^2 + 1. $$ Solve for \(\lambda\) to find the eigenvalues: $$ \lambda^2 + 1 = 0 \Rightarrow \lambda = \pm i. $$

Determine stability based on eigenvalues

Now that we have the eigenvalues of the Jacobian, we can use their properties to determine the stability of the system. Recall that a fixed point is: 1. Stable (asymptotically stable) if all eigenvalues have negative real parts. 2. Unstable if at least one eigenvalue has a positive real part. 3. Semi-stable if there are eigenvalues with real parts equal to zero and no eigenvalues with positive real parts. In this case, our eigenvalues are purely imaginary: \(\pm i\), which means their real parts are zero. Therefore, the fixed point (0,0) is semi-stable.

Describe the relationship between f(x, y) and the stability

The function f(x, y) is analytic at the origin and f(0,0)=0, which has been used to linearize the system of differential equations. However, as we have found, the stability type is independent of the function f(x, y) since the eigenvalues are purely imaginary, and the fixed point (0,0) is semi-stable regardless of the actual behavior of f(x, y). In conclusion, the type of stability of this system (semi-stable) is not influenced by the function f(x, y) at the origin.

Key concepts

Jacobian Matrix

In the realm of differential equations, the Jacobian Matrix is a vital tool for examining the local behavior of a system near a point. Specifically, when dealing with a set of non-linear differential equations, the Jacobian Matrix consists of the first partial derivatives of the system’s function with respect to its variables.

Considering the given exercise, we look at a two-dimensional system with variables x and y. The Jacobian Matrix plays a crucial role in linearizing the system around the origin. When we evaluate this matrix at a fixed point, in this case, the origin \( (0,0) \), we can analyze the system's behavior near that point. By doing this, we can transition from the complex landscape of the non-linear function to the more straightforward dynamics of a linear system, which can be easier to solve and understand.

Understanding the role of the Jacobian Matrix in stability analysis is key for students. Not only does it help to simplify the problem, but provides a bridge to predict the behavior of systems in more general situations, just by examining their properties at specific points.

Eigenvalues of Differential Equations

Moving deeper into the study of differential equations, the eigenvalues of a system have profound implications on the system's stability. Derived from the Jacobian Matrix, eigenvalues provide insight into the qualitative behavior of a system at equilibrium points.

For example, in our exercise, we solved the characteristic polynomial of the Jacobian Matrix at the origin to find the eigenvalues of the system. These eigenvalues, represented as \( \pm i \), are purely imaginary numbers. This specific quality of having zero real parts implies that the system is not attracted to or repelled from the origin—hence, why the origin is denoted semi-stable.

Students should grasp the significance of eigenvalues beyond calculation. They should understand the link between the sign and complexity of these values and the system's behavior. Stability, oscillations, and even chaos in dynamic systems can be foreseen just by studying the eigenvalues, which can act as a compass to navigate the complex behavior of differential systems.

Linearization of Differential Systems

Linearization is a pivotal technique that simplifies the analysis of differential systems by approximating them near a point of interest. This method is especially relevant when examining the stability of non-linear systems. By creating a linear model that closely represents the system's behavior around an equilibrium point, we can predict the system’s local dynamics.

The linearization process uses the Jacobian Matrix at a specific point, commonly an equilibrium or fixed point, to create a linear system whose solution gives insight into the stability of the non-linear system. In our exercise, we linearized around the origin (0,0), leading to a linear system whose stability type could then be discerned through the Jacobian's eigenvalues.

It is important for students to recognize that linearization is an approximation. While it provides valuable information about the immediate vicinity of the fixed point, it does not necessarily reflect the global behavior of the system. However, it is a powerful first step in understanding complex dynamics.

Analytic Functions

Analytic functions, often encountered in complex analysis and differential equations, are functions that can be locally expressed by a convergent power series. In our context, the function \( f(x, y) \) is required to be analytic at the origin, meaning we can expand it into a power series around that point. The analyticity of \( f \) ensures that the function's behavior is smooth and predictable near the origin, which, in turn, allows us to use linearization techniques reliably.

In the exercise, the function \( f \) also passes through the origin, with \( f(0,0)=0 \). This assures that the linearized model is a valid approximation around (0,0). Even though \( f \) itself does not affect the semi-stable classification in our case, the property of analyticity is crucial for the stability analysis in other scenarios. By valuing analytic functions in differential equations, students will find a direct application of advanced calculus concepts in understanding physical phenomena and systems in engineering, physics, and beyond.