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Chapter 9: Problem 3

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. $$ d x / d t=(1+x) \sin y, \quad d y / d t=1-x-\cos y $$

Short Answer

Expert verified
Answer: The critical point (0,0) is a center and its stability is neutral.

Step by step solution

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01

Verify the critical point (0, 0)

To find the critical points, set both rates of change equal to 0 and solve for x and y: $$ \begin{aligned} (1+x) \sin y &= 0 \\ 1-x-\cos y &= 0 \end{aligned} $$ In this case, it's easy to see that \((0,0)\) is a critical point, as \(\sin 0 = 0\) and \(1 - 0 - \cos 0 = 1 - 1 = 0\).
02

Find the Jacobian matrix of the system

The Jacobian matrix of the system of equations is a matrix of partial derivatives, which can be written as: $$ J(x,y) = \begin{bmatrix} \frac{\partial {(1 + x)\sin y}}{\partial x} & \frac{\partial {(1 + x)\sin y}}{\partial y} \\ \frac{\partial {(1 - x - \cos y)}}{\partial x} & \frac{\partial {(1 - x - \cos y)}}{\partial y} \end{bmatrix} $$ Compute the partial derivatives: $$ J(x,y) = \begin{bmatrix} \sin y & (1+x)\cos y \\ -1 & \sin y \end{bmatrix} $$
03

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

Substitute the critical point \((0,0)\) into the Jacobian matrix: $$ J(0,0) = \begin{bmatrix} \sin 0 & (1+0)\cos 0 \\ -1 & \sin 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $$
04

Find the eigenvalues of the Jacobian matrix at the critical point

Find the eigenvalues of the evaluated Jacobian matrix by first calculating its characteristic polynomial: $$\begin{aligned} \det(J(0,0)-\lambda I) &= \det \begin{bmatrix} -\lambda & 1 \\ -1 & -\lambda \end{bmatrix} = \lambda^2 + 1 \end{aligned} $$ Since the characteristic polynomial is \(\lambda^2 + 1 = 0\), this equation has complex eigenvalues with nonzero imaginary parts, \(\lambda = \pm i\). Since the real parts of the eigenvalues are zero, the critical point \((0,0)\) is a center, which is an almost-linear system.
05

Determine the type and stability of the critical point

As the eigenvalues are purely imaginary, the critical point \((0,0)\) corresponds to a center. In general, a center is considered neutrally stable, as it is not attracting or repelling. Hence, the type of the critical point \((0,0)\) is a center and the stability of the critical point is neutral.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Jacobian Matrix Differential Equations
The Jacobian matrix is a powerful tool used to analyze the behavior of differential equations near critical points. It consists of first-order partial derivatives of each function in a system with respect to all variables.

In the context of the given system of equations, calculating the Jacobian matrix helps us linearize the system near the critical point \( (0,0) \). When we find this matrix at a critical point, it represents the best linear approximation to the system's dynamics near that point. In our case, the Jacobian matrix evaluated at the critical point \( (0,0) \) turns out to be a rotation matrix. This implies that the system exhibits rotational behavior around the critical point.

Understanding the Jacobian is crucial because it directly influences the determination of the critical point's stability - a key factor in predicting the system's long-term behavior.
Eigenvalues of a Matrix
Eigenvalues are a fundamental concept when assessing the stability of a system. They are values that, when a certain matrix is multiplied by a vector, the vector is only scaled and not rotated. To find eigenvalues, we set up and solve the characteristic equation of the matrix.

In the provided exercise, the eigenvalues of the Jacobian matrix are found by calculating the roots of the characteristic polynomial \( \lambda^2 + 1 \). The roots are complex numbers \( \lambda = \pm i \), suggesting oscillatory motion without growth or decay since the real part of the eigenvalues is zero. These complex eigenvalues are keys to determining that our system's behavior near the critical point is that of a neutral center.

Knowing the eigenvalues helps us understand whether perturbations to the system will damp out, persist undamped, or grow exponentially, which speaks directly to system stability.
Neutral Stability Center
A neutral stability center is a particular type of critical point in a dynamical system. It is characterized by undamped periodic motion. In simpler terms, any small push or pull on the system will lead to it moving in a loop without spiraling inwards or outwards.

In our exercise, the term 'neutral stability' is derived from the purely imaginary eigenvalues of the Jacobian matrix at the critical point. This type of critical point neither attracts nearby points (like a spiral) nor repels them. Instead, trajectories are confined to close curves around the critical point, indicating a pattern of constant amplitude oscillations. A good real-world analogy would be a marble freely rotating inside a smooth bowl; it continues circling around the bottom without settling or escaping.

Navigating through these concepts can be tough, but unpacking the role of the Jacobian matrix, eigenvalues, and the type of critical point can provide a significant leap in understanding the stability of systems governed by differential equations.

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Most popular questions from this chapter

The equation $$ u^{\prime \prime}-\mu\left(1-\frac{1}{3} u^{\prime 2}\right) u^{\prime}+u=0 $$ is often called the Rayleigh equation. (a) Write the Rayleigh equation as a system of two first order equations. (b) Show that the origin is the only critical point of this system. Determine its type and whether it is stable or unstable. (c) Let \(\mu=1 .\) Choose initial conditions and compute the corresponding solution of the system on an interval such as \(0 \leq t \leq 20\) or longer. Plot \(u\) versus \(t\) and also plot the trajectory in the phase plane. Observe that the trajectory approaches a closed curve (limit cycle). Estimate the amplitude \(A\) and the period \(T\) of the limit cycle. (d) Repeat part (c) for other values of \(\mu,\) such as \(\mu=0.2,0.5,2,\) and \(5 .\) In each case estimate the amplitude \(A\) and the period \(T\). (e) Describe how the limit cycle changes as \(\mu\) increases. For example, make a table of values and/or plot \(A\) and \(T\) as functions of \(\mu .\)

As mentioned in the text, one improvement in the predator-prey model is to modify the equation for the prey so that it has the form of a logistic equation in the absence of the predator. Thus in place of Eqs. ( 1 ) we consider the system $$ d x / d t=x(a-\sigma x-\alpha y), \quad d y / d t=y(-c+\gamma x) $$ where \(a, \sigma, \alpha, c,\) and \(\gamma\) are positive constants. Determine all critical points and discuss their nature and stability characteristics. Assume that \(a / \sigma \gg c / \gamma .\) What happens for initial data \(x \neq 0, y \neq 0 ?\)

(a) Find an equation of the form \(H(x, y)=c\) satisfied by the trajectories. (b) Plot several level curves of the function \(H\). These are trajectories of the given system. Indicate the direction of motion on each trajectory. $$ \text { Duffing's equation: } \quad d x / d t=y, \quad d y / d t=-x+\left(x^{3} / 6\right) $$

The equation of motion of an undamped pendulum is \(d^{2} \theta / d t^{2}+\omega^{2} \sin \theta=0,\) where \(\omega^{2}=g / L .\) Let \(x=\theta, y=d \theta / d t\) to obtain the system of equations $$ d x / d t=y, \quad d y / d t=-\omega^{2} \sin x $$ (a) Show that the critical points are \((\pm n \pi, 0), n=0,1,2, \ldots,\) and that the system is almost lincar in the neighborhood of cach critical point. (b) Show that the critical point \((0,0)\) is a (stable) center of the corresponding linear system. Using Theorem 9.3.2 what can be said about the nonlinear system? The situation is similar at the critical points \((\pm 2 n \pi, 0), n=1,2,3, \ldots\) What is the physical interpretation of these critical points? (c) Show that the critical point \((\pi, 0)\) is an (unstable) saddle point of the corresponding linear system. What conclusion can you draw about the nonlinear system? The situation is similar at the critical points \([\pm(2 n-1) \pi, 0], n=1,2,3, \ldots\) What is the physical interpretation of these critical points? (d) Choose a value for \(\omega^{2}\) and plot a few trajectories of the nonlinear system in the neighborhood of the origin. Can you now draw any further conclusion about the nature of the critical point at \((0,0)\) for the nonlinear system? (e) Using the value of \(\omega^{2}\) from part (d) draw a phase portrait for the pendulum. Compare your plot with Figure 9.3 .5 for the damped pendulum.

(a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and portrait for the system. (c) From the plot(s) in part (b) determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. $$ d x / d t=-x+2 x y, \quad d y / d t=y-x^{2}-y^{2} $$
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