Section 5.7. Classify Phase planes

Objective:

Classify Phase planes with eigenvalues

In this section, we are going to summarize all cases that we considered and classify all phase planes. Recall that a given matrix $A$ and a system of homogeneous differential equation $\overrightarrow{X}=A\overrightarrow{x}$, we can find the eigenvalues of the matrix $A$ and then eigenvectors. $\text{det}(A-\lambda I)=0=f(\lambda),$ the characteristics equation determinate the eigenvalues. 

Case 1: When $f(\lambda)=0$ has two distinct positive real solutions, $\lambda_{1} \gt \lambda_{2} \gt 0$. In this case, we have two eigenvectors, $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$. The entries of $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=c_{1}e^{\lambda_{1}t}\overrightarrow{v_{1}}+c_{2}e^{\lambda_{2}t}\overrightarrow{v_{2}}$. The phase plane has two straight lines such that they present the eigenvector directions. The arrows are pointing away from the $(0,0)$. The $\overrightarrow{x}=\overrightarrow{0}$ is called a nodal source and is an unstable solution. 

Case 2: When $f(\lambda)=0$ has two distinct negative real solutions, $\lambda_{1} \lt \lambda_{2} \lt 0$. }In this case, we have two eigenvectors, $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$. The entries of $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=c_{1}e^{\lambda_{1}t}\overrightarrow{v_{1}}+c_{2}e^{\lambda_{2}t}\overrightarrow{v_{2}}$. The phase plane has two straight lines such that they present the eigenvector directions. The arrows are pointing toward the $(0,0)$. The $\overrightarrow{x}=\overrightarrow{0}$ is called a nodal sink and is a stable solution. 

Case 3: When $f(\lambda)=0$ has two distinct real solutions, one is positive and one is negative, $\lambda_{1} \lt 0 \lt \lambda_{2}$}. In this case, we have two eigenvectors, $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$. The entries of $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=c_{1}e^{\lambda_{1}t}\overrightarrow{v_{1}}+c_{2}e^{\lambda_{2}t}\overrightarrow{v_{2}}$. The phase plane has two straight lines such that they present the eigenvector directions. The arrow on $\overrightarrow{v_{1}}$ is pointing toward the $(0,0)$, and the arrow on $\overrightarrow{v_{2}}$ is pointing away from $(0,0)$. The $\overrightarrow{x}=\overrightarrow{0}$ is called a saddle point and is an unstable solution.

Example 1: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -4 & -4\\ 0 & -1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Exercise 1: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -2 & 0\\ 3 & 1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Case 4: When $f(\lambda)=0$ has two distinct complex solutions, $\lambda=a\pm bi$ and $a\neq0$. In this case, we have two eigenvectors, $\overrightarrow{v_{1}}=\overrightarrow{p}+i\overrightarrow{q}$ and $\overrightarrow{v_{2}}=\overrightarrow{p}-i\overrightarrow{q}$. The entries of $\overrightarrow{p}$ and $\overrightarrow{q}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=e^{at}(c_{1}(\text{cos}(bt)\overrightarrow{p}-\text{sin}(bt)\overrightarrow{q})+c_{2}(\text{sin}(bt)\overrightarrow{p}+\text{cos}(bt)\overrightarrow{q}))$. The entries of the vector $\overrightarrow{x(t)}$ have $\text{cos}(bt)$ and $\text{sin}(bt)$. The phase plane has clock-wise spiral orientations. The arrows on are pointing toward the $(0,0)$ if $a \lt 0$, and the arrows are pointing away from $(0,0)$ if $a \gt 0$. The $\overrightarrow{x}=\overrightarrow{0}$ is called a spiral sink and is a stable solution for $a \lt 0$ case, and is called a spiral source and is an unstable solution for $a \gt 0$ case.

Example 2: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -1 & 2\\ -2 & -1 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Exercise 2: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & -3\\ 3 & 2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Case 5: When $f(\lambda)=0$ has  two distinct complex solutions, $\lambda=\pm bi$.} In this case, we have two eigenvectors, $\overrightarrow{v_{1}}=\overrightarrow{p}+i\overrightarrow{q}$ and $\overrightarrow{v_{2}}=\overrightarrow{p}-i\overrightarrow{q}$. The general solution will be $\overrightarrow{x(t)}=(c_{1}(\text{cos}(bt)\overrightarrow{p}-\text{sin}(bt)\overrightarrow{q})+c_{2}(\text{sin}(bt)\overrightarrow{p}+\text{cos}(bt)\overrightarrow{q}))$. The entries of the vector $\overrightarrow{x(t)}$ have $\text{cos}(bt)$ and $\text{sin}(bt)$. The phase plane has counter-clock-wise or clock-wise circular ( ellipse) orientations. The arrows counter-clock-wise or counter-clock-wise. The $\overrightarrow{x}=\overrightarrow{0}$ is called center and is a (neutrally) stable solution. 

Example 3: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & -4\\ 2 & -2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Exercise 3: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -3 & -5\\ 2 & 3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Case 6: When $f(\lambda)=0$ has one real solution, $\lambda_{1}$, and two eigenvectors, $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$. }The entries of $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=c_{1}e^{\lambda_{1}t}\overrightarrow{v_{1}}+c_{2}e^{\lambda_{1}t}\overrightarrow{v_{2}}$. The phase plane has straight lines passing $(0,0)$. The arrow those lines are pointing toward the $(0,0)$ if $\lambda_{1} \lt 0$, and the arrows are pointing away from $(0,0)$ if $\lambda_{1} \gt 0$. The $\overrightarrow{x}=\overrightarrow{0}$ is called a star node, sometimes is called proper node. It is a stable solution if [latex]\lambda_{1} \lt 0,[/latex] and an unstable solution if $\lambda_{1} \gt 0$.

Example 4: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & 0\\ 0 & 2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Exercise 4: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -3 & 0\\ 0 & -3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Case 7: When $f(\lambda)=0$ has one real solution, $\lambda_{1}$, and one eigenvector, $\overrightarrow{v}$. The entries of $\overrightarrow{v}$ are real numbers. The general solution will be $\overrightarrow{x(t)}=c_{1}e^{\lambda_{1}t}\overrightarrow{v}+c_{2}[e^{\lambda_{1}t}\overrightarrow{w}+e^{\lambda_{1}t}t\overrightarrow{v}]$ where $(A-\lambda_{1})\overrightarrow{w}=\overrightarrow{v}$. The phase plane has one straight line passing $(0,0)$. The arrow those lines are pointing toward the $(0,0)$ if $\lambda_{1} \lt 0$, and the arrows are pointing away from $(0,0)$ if $\lambda_{1} \gt 0$. The $\overrightarrow{x}=\overrightarrow{0}$ is called an improper node. It is a stable solution if $\lambda_{1} \lt 0,$ and an unstable solution if $\lambda_{1} \gt 0$.

Example 5: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} 2 & 3\\ 0 & 2 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is. 

Exercise 5: Given $\left[\begin{array}{c} x_{1}'\\ x_{2}' \end{array}\right]=\left[\begin{array}{cc} -3 & 0\\ 2 & -3 \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right]=A\overrightarrow{x}$, find the general solution of the system of equations. Classify the stability of the equilibrium solution $\overrightarrow{x}=\overrightarrow{0}$ and describe what kind of critical point it is.