Question

(a) Rewrite the given second order equation as an equivalent first order system. (b) Graph the nullclines of the autonomous system and locate all equilibrium points. (c) As in Figure 6.6, sketch direction field arrows on the nullclines. Also, sketch an arrow in each open region that suggests the direction in which a solution point is moving when it is in that region. $$ y^{\prime \prime}+y+y^{3}=0 $$

Short answer

#Answer# The second-order ODE \(y^{\prime \prime} + y + y^3 = 0\) can be rewritten as an equivalent first-order system: $$\begin{cases} x_1^{\prime} = x_2 \\ x_2^{\prime} = -x_1 - x_1^3 \end{cases}$$ The nullclines are \(x_1^\prime = x_2 = 0\) and \(x_2^\prime = -x_1 - x_1^3 = 0\). There is only one equilibrium point, \((0,0)\). On the horizontal nullcline, arrows point downward if \(x_1>0\) and upward if \(x_1<0\). On the second nullcline, arrows point to the right if \(x_2>0\) and to the left if \(x_2<0\). In the open regions, arrows point in the following directions: bottom-right if \(x_1>0\) and \(x_2>0\), top-right if \(x_1<0\) and \(x_2>0\), top-left if \(x_1<0\) and \(x_2<0\), and bottom-left if \(x_1>0\) and \(x_2<0\).

Step-by-step solution

Converting Second Order ODE into First Order System

We are given the second order ODE: \(y^{\prime \prime} + y + y^3 = 0\). Let's introduce another variable, \(x_1=y\), \(x_2=y^{\prime}\), and rewrite the second-order equation as a system of two first-order ODEs: $$ \begin{cases} x_1^{\prime} = y^{\prime} = x_2 \\ x_2^{\prime} = y^{\prime \prime} = -x_1 - x_1^3 \end{cases} $$ Now we have an equivalent first-order system.

Finding the nullclines

The nullclines are the curves where \(x_1^\prime = 0\) or \(x_2^\prime = 0\). Solving these equations, we get: $$ \begin{cases} x_1^\prime = x_2 = 0 \\ x_2^\prime = -x_1 - x_1^3 = 0 \end{cases} $$ The first nullcline is the horizontal line \(x_2=0\). The second nullcline is the curve \(-x_1-x_1^3=0\).

Locate equilibrium points

An equilibrium point occurs where both \(x_1^{\prime} = 0\) and \(x_2^{\prime} = 0\). We already have these equations from step 2. Solving these equations simultaneously, we get: $$ \begin{cases} x_2 = 0 \\ -x_1 - x_1^3 = 0 \end{cases} $$ We have \(x_1(1+x_1^2) = 0\), which gives us \(x_1=0\) and \(x_1^2 = -1\). There is only one equilibrium point, \((0,0)\).

Sketching direction field arrows on nullclines and in open regions

To sketch the direction field arrows, we need to examine the signs of \(x_1^{\prime}\) and \(x_2^{\prime}\) on each nullcline and in each open region. (a) On the nullcline \(x_1^{\prime} = 0\), we have \(x_2 = 0\). Therefore, in this case, \(x_1^{\prime} = 0\) and \(x_2^{\prime} = -x_1- x_1^3\). - When \(x_1>0\), \(x_2^{\prime}<0\), and hence the arrows will point downward. - When \(x_1<0\), \(x_2^{\prime}>0\), and hence the arrows will point upward. (b) On the nullcline \(x_2^{\prime} = 0\), we have \(-x_1-x_1^3 = 0\). Therefore, in this case, \(x_1^{\prime} = x_2\) and \(x_2^{\prime}=0\). - Since \(x_1^{\prime} = x_2\), when \(x_2>0\), the arrows will point to the right. - When \(x_2<0\), the arrows will point to the left. (c) In the open regions, we analyze the signs of \(x_1^{\prime}\) and \(x_2^{\prime}\). - If \(x_1>0\) and \(x_2>0\), we'll have \(x_1^{\prime}>0\) and \(x_2^{\prime}<0\), so the arrows will point towards the bottom-right. - If \(x_1<0\) and \(x_2>0\), we'll have \(x_1^{\prime}>0\) and \(x_2^{\prime}>0\), so the arrows will point towards the top-right. - If \(x_1<0\) and \(x_2<0\), we'll have \(x_1^{\prime}<0\) and \(x_2^{\prime}>0\), so the arrows will point towards the top-left. - If \(x_1>0\) and \(x_2<0\), we'll have \(x_1^{\prime}<0\) and \(x_2^{\prime}<0\), so the arrows will point towards the bottom-left. Using this information, a direction field plot can be sketched with arrows indicating the direction of solutions in various regions of the phase plane.

Key concepts

Nullclines

Nullclines are significant to understanding the behavior of dynamic systems. They are curves on which the derivative of the system with respect to one of the variables is zero. In the context of a first order system converted from a second order ODE, nullclines help to visually identify how solutions behave with respect to each variable.

Specifically, in the provided exercise, nullclines divide the phase plane into regions where the sign of the derivatives changes. This sign change indicates the direction in which solutions move. When analyzing nullclines, it's important to recognize that they provide boundaries – solutions cannot cross them in the direction where the derivative is zero but must instead move along or away from them.

For the exercise given, when plotting the nullclines, one can immediately distinguish between different behaviors of the dynamic system, as these curves are essentially the 'skeleton' of the direction field.

Equilibrium Points

An equilibrium point of a differential equation is a point where all derivatives are zero, indicating that the system is at rest. It is a cornerstone concept in the analysis of differential equations, as it signifies a state where no change occurs over time in the system's variables.

To find equilibrium points, as done in step 3 of the solution, one must solve for the points where the derivatives of all system equations are zero simultaneously. These points usually represent the long-term behavior of the system. In our exercise, we have an equilibrium point at \( (0,0) \). This particular point is where the nullclines intersect, highlighting its importance as a fulcrum in the direction field, around which the system's behavior is organized.

Direction Field

Direction fields, also known as vector fields or phase portraits, are visual representations of the solutions of a first order system of differential equations. They consist of arrows indicating the direction in which the solution curves move at any given point on the plane.

Creating a direction field for the system enables us to predict the qualitative behavior of solutions without solving the equations analytically. This graphical tool is particularly helpful for understanding the structure of solutions in the vicinity of equilibrium points. In step 4 of the provided solution, the direction of arrows on nullclines and in the regions between them provides insight into the stability of the equilibrium point and the general flow of the system.

First Order ODEs

First order ordinary differential equations (ODEs) form the basis of our analysis of more complex, higher-order systems. A first order ODE involves derivatives of only a single variable and can be represented graphically using direction fields as we have discussed.

By converting a second order ODE to a system of first order ODEs, as in step 1 of the exercise, we enable the use of graphical methods such as nullclines and direction fields to analyze the system dynamics. First order systems are essential as they simplify complex dynamics into manageable parts and are often easier to solve both analytically and numerically. The study of first order ODEs is a fundamental step toward understanding multi-dimensional systems in advanced mathematics.