Question

Verify that \((0,0)\) is a critical point, show that the system is almost linear, and discuss the type and stability of the critical point \((0,0)\) by examining the corresponding linear system. $$ d x / d t=x-y^{2}, \quad d y / d t=x-2 y+x^{2} $$

Short answer

In conclusion, we have shown that the critical point \((0,0)\) is indeed a critical point of the given system. We found the Jacobian matrix and evaluated it at the critical point to determine the type and stability, revealing that it is an unstable saddle point. Furthermore, by comparing the original nonlinear system with its linear approximation given by the Jacobian matrix, we demonstrated that the system is almost linear near the critical point as the nonlinear terms are small when \((x,y)\) is close to \((0,0)\).

Step-by-step solution

Verify the critical point \((0,0)\).

To verify that \((0,0)\) is a critical point, we need to find the points where both \(dx/dt\) and \(dy/dt\) are equal to zero. In this case, we have the following equations for \(dx/dt\) and \(dy/dt\): $$ \frac{dx}{dt} = x - y^2, \quad \frac{dy}{dt} = x - 2y + x^2 $$ Setting these equations to zero and solving for x and y, we can see that \((0,0)\) is indeed a critical point.

Find the Jacobian matrix.

The Jacobian matrix for the given system is computed by taking the partial derivatives of each equation with respect to each variable. In this case, we have: $$ J(x,y) = \begin{bmatrix} \frac{\partial (x - y^2)}{\partial x} & \frac{\partial (x - y^2)}{\partial y} \\ \frac{\partial (x - 2y + x^2)}{\partial x} & \frac{\partial (x - 2y + x^2)}{\partial y} \end{bmatrix} = \begin{bmatrix} 1 & -2y \\ 1+2x & -2 \end{bmatrix} $$

Evaluate the Jacobian matrix at the critical point\((0,0)\).

In order to examine the type and stability of the critical point \((0,0)\), we need to evaluate the Jacobian matrix at this point: $$ J(0,0) = \begin{bmatrix} 1 & 0 \\ 1 & -2 \end{bmatrix} $$

Discuss the type and stability of the critical point \((0,0)\).

We will analyze the eigenvalues of the Jacobian matrix evaluated at the critical point \((0,0)\) to determine the type and stability: The eigenvalues of \(J(0,0)\) are the solutions of the following characteristic equation: $$\det(J(0,0) - \lambda I) = \begin{vmatrix} 1 - \lambda & 0 \\ 1 & -2 - \lambda \end{vmatrix} = (1-\lambda)(-2-\lambda) $$ The eigenvalues are \(\lambda_1 = 1\) and \(\lambda_2 = -2\). Since one of the eigenvalues is positive, and the other is negative, the critical point \((0,0)\) is a saddle point, which implies it is an unstable equilibrium.

Show the system is almost linear.

In order to show the system is almost linear, we compare the original nonlinear system with its linear approximation given by the Jacobian matrix. The linear approximation is given by: $$ \begin{bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix} = J(0,0) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ x - 2y \end{bmatrix} $$ Comparing the linear approximation with the original system, we observe that the only nonlinear terms are \(y^2\) and \(x^2\). Since these terms are small when \((x,y)\) is close to the critical point \((0,0)\), the behavior of the system will be similar to the linear approximation near the equilibrium point, which implies that the system is almost linear in the neighborhood of \((0,0)\).

Key concepts

Jacobian matrix

The Jacobian matrix is a cornerstone of critical point analysis in nonlinear dynamical systems.

It consists of first-order partial derivatives of a vector function. Specifically, when dealing with a dynamical system described by differential equations, the Jacobian helps us linearize the system near a given point, which is often a critical (equilibrium) point.

The Jacobian matrix for a two-dimensional system, like our current example, will be a 2x2 matrix where each entry is the derivative of the system's output with respect to its input variables. At the critical point \(0,0\), the Jacobian matrix provides a linear approximation of the system's dynamics.

For better understanding, one should closely examine how it represents the rate of change of the system when moving away from the equilibrium. It is this local linear approximation that enables further analysis, such as determining stability through eigenvalues.

Eigenvalues

Eigenvalues are intrinsic values associated with a matrix that provide significant insights into the matrix's properties and, by extension, the dynamics of a system described by that matrix.

To find these values for the Jacobian matrix, we compute the characteristic equation and solve for \(\lambda\). The signs and magnitudes of the eigenvalues play a critical role in determining system behavior at a critical point.

In our exercise, with \(\lambda_1 = 1\) and \(\lambda_2 = -2\), we are given contrasting bits of information: a positive eigenvalue indicates a direction of instability (since solutions grow exponentially), while a negative one suggests stability (with solutions decaying over time). The combination signifies a saddle point — an unstable equilibrium where different trajectories behave differently based on their approach direction.

Linear stability

Linear stability analysis helps determine the future behavior of a dynamical system that is near an equilibrium point.

It involves analyzing the linearized version of the system, which in our case is derived from the Jacobian matrix. By studying the eigenvalues of the Jacobian matrix evaluated at the critical point, we can infer the type of equilibrium: whether it’s stable, unstable, or otherwise.

A stable critical point would have all negative eigenvalues, implying that trajectories would converge to the point. Conversely, one positive eigenvalue is enough to declare the system unstable, as is the situation in our exercise. It’s important to note that for higher-dimensional systems, the analysis can become more complex with the introduction of complex eigenvalues.

Nonlinear dynamical systems

Our exercise deals with a nonlinear dynamical system, characterized by its nonlinear terms (such as \(y^2\) and \(x^2\)). These terms introduce rich and sometimes complex behavior that cannot be fully unveiled through linear methods alone.

Critical point analysis provides a window into understanding such systems by allowing us to approximate their behavior near equilibrium points. This local view is essential, particularly in cases where an analytical solution to the nonlinear system is not attainable.

However, it is paramount to remember that the analysis we conduct through the Jacobian matrix and eigenvalues only holds true in the neighborhood of the critical point, as nonlinear behavior may differ significantly further away. A system being 'almost linear' means that the nonlinear elements have minor influence near the equilibrium, but their impact could grow with distance from the critical point.