Definition 179.
A solution \(\mathbf{x}(t)\) to a system of differential equations is called a trajectory.
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Handout Lesson 28, Pictures of Solution Curves of Linear Systems
Textbook Section(s).
This lesson is based on Section 5.3 of your textbook by Edwards, Penney, and Calvis.
Categorizing Solutions for \(\mathbf{x}'=A\mathbf{x}\) when \(A\) is a \(2\times 2\) matrix.
When \(A\) is a \(2\times 2\) matrix, there are only three different types of solutions that \(\mathbf{x}'=A\mathbf{x}\) can have based on its eigenvalues and eigenvectors.
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\(\displaystyle \mathbf{x}(t)=c_1e^{\lambda_1 t}\mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\)
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\(A\) has two distinct real eigenvalues (\(\lambda_1\) and \(\lambda_2\) with eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.)
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\(A\) has only one real eigenvalue (\(\lambda_1=\lambda_2\)) that yields two linearly independent eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{.}\)
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\(\displaystyle \mathbf{x}(t)=c_1e^{pt}(\mathbf{a}\cos(qt)- \mathbf{b}\sin(qt)) +c_2e^{pt}(\mathbf{a}\sin(qt)+ \mathbf{b}\cos(qt))\)
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\(A\) has a complex conjugate pair of eigenvalues \(p \pm qi\) with eigenvectors \(\mathbf{a}\pm i \mathbf{b}\text{.}\)
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\(\displaystyle \mathbf{x}(t)=c_1e^{\lambda t}\mathbf{v_1}+ c_2e^{\lambda t}(t\mathbf{v_1}+\mathbf{v_2})\)
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\(A\) has only one real eigenvalue (\(\lambda=\lambda_1=\lambda_2\)), and it only corresponds to one linearly independent eigenvector.
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Recall from Section 5.5 that \(\mathbf{v_1}\) is an eigenvector of \(A\)\begin{gather*} (A-\lambda I)\mathbf{v_1}=\mathbf{0} \end{gather*}and \(\mathbf{v_2}\) is nonzero vector that is a solution of\begin{gather*} (A-\lambda I)\mathbf{v_2}= \mathbf{v_1} \end{gather*}is also a nonzero vector (and, hence, an eigenvector of \(A\) for eigenvalue \(\lambda\text{.}\))
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The solution curves in \(x_1x_2\)-space are dependent on the eigenvalues and eigenvectors. We will look at a few of these cases in depth. If we are not able to look at all of the cases, then I will refer you to your textbook.
Please note that unless otherwise stated, all of the pictures in this set of notes were created using Sagemath. Section 5.3 of your textbook by Edwards, Penney, and Calvis also has many helpful pictures.
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\).
Case 1: \(A\) has two distinct real eigenvalues of opposite signs (\(\lambda_1
> 0\) and \(\lambda_2 < 0\) with eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.)
This case produces a saddle point. In the picture below, \(\lambda_1>0\) and \(\lambda_2<0\text{.}\) The picture displays a vector field for a system with a positive eigenvalue and a negative eigenvalue. The eigenvectors are included and several solution curves. The solution curves move away from the line associated with the eigenvector for the negative eigenvalue and toward the line associated with the eigenvector for the positive eigenvalue.
Vector field for a system with a positive eigenvalue and a negative eigenvalue
Some Definitions:.
Before we continue, we introduce a few definitions.
A trajectory can be seen as a set of parametric equations describing a curve. Curves defined by parametric equations have directions, so we can place arrows on trajectories indicating how they are traversed as \(t\) increases.
Definition 180.
The origin is called a node of the system \(\mathbf{x}'=A\mathbf{x}\) if:
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Every trajectory either
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flows into the origin, \(\left(\lim_{t\rightarrow \infty} (x_1(t),x_2(t))=(0,0)\right)\text{,}\) sink, OR
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every trajectory flows away from the origin, \(\left(\lim_{t\rightarrow \infty} \sqrt{(x_1(t))^2+(x_2(t))^2}=\infty\right)\text{,}\) source
AND -
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Every trajectory has a tangent line at the origin.
I also want to note that your textbook says that a node is proper if “no two different pairs of ’opposite’ trajectories are tangent to the same straight line through the origin” and it is improper otherwise. Consequently, if at least four trajectories have the same tangent line at a node, then the node must be improper.
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 2: \(A\) has two distinct positive eigenvalues (\(\lambda_1\) and \(\lambda_2\) with eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.)
This case produces an improper nodal source. In the picture below, \(0<\lambda_2<\lambda_1\text{.}\) The picture displays a vector field for a system with two distinct positive eigenvalues The eigenvectors are included and several solution curves. The solution curves move away from the line associated with the eigenvector for the smaller eigenvalue and become more parallel to the line associated with the eigenvector for the larger eigenvalue.
Vector field for a system with two distinct positive eigenvalues
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 3: \(A\) has two distinct negative eigenvalues (\(\lambda_1\) and \(\lambda_2\) with eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.)
This case produces an improper nodal sink. In the picture below, \(\lambda_1<\lambda_2<0\text{.}\) The picture displays a vector field for a system with two distinct negative eigenvalues The eigenvectors are included and several solution curves. The solution curves start out nearly parallel to the line associated with the eigenvector for the smaller eigenvalue (larger in magnitude) and approach the line associated with the eigenvector for the larger eigenvalue (of smaller magnitude).
Vector field for a system with two distinct negative eigenvalues
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 4: \(A\) has a negative eigenvalue (\(\lambda_1\)) and an eigenvalue equal to zero (\(\lambda_2\)). The eigenvectors are \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.
Your book identifies these as parallel lines. In the picture below, \(\lambda_1< 0\) and \(\lambda_2=0\text{.}\) The picture displays a vector field for a system with a negative eigenvalue and a zero eigenvalue. The eigenvectors are included and several solution curves. The solution curves are parallel to the line associated with the eigenvector for the negative eigenvalue and approach the line associated with the eigenvector for the zero eigenvalue.
Vector field for a system with one negative eigenvalue and one zero eigenvalue
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 5: \(A\) has a positive eigenvalue (\(\lambda_1\)) and an eigenvalue equal to zero (\(\lambda_2\)). The eigenvectors are \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{,}\) respectively.
Your book identifies these as parallel lines. In the picture below, \(\lambda_1> 0\) and \(\lambda_2=0\text{.}\) The picture displays a vector field for a system with a positive eigenvalue and a zero eigenvalue. The eigenvectors are included and several solution curves. The solution curves are parallel to the line associated with the eigenvector for the positive eigenvalue and approach the line associated with the eigenvector for the zero eigenvalue.
Vector field for a system with one positive eigenvalue and one zero eigenvalue
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 6: \(A\) has a positive eigenvalue of multiplicity 2 (\(\lambda_1 = \lambda_2 > 0\)) that corresponds to two linearly independent eigenvectors are \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{.}\)
This case produces a proper nodal source. In the picture below, \(\lambda_1 = \lambda_2>0\text{.}\) The picture displays a vector field for a system with a single positive eigenvalue that corresponds to two linearly independent eigenvectors. Several solution curves are shown. The solution curves are rays emanating from the origin.
Vector field for a system with one positive eigenvalue of multiplicity 2
Solution curves of the form \(\mathbf{x}(t)=c_1e^{\lambda_1 t} \mathbf{v_1}+ c_2e^{\lambda_2 t}\mathbf{v_2}\) continued.
Case 7: \(A\) has a negative eigenvalue of multiplicity 2 (\(\lambda_1 = \lambda_2 < 0\)) that corresponds to two linearly independent eigenvectors are \(\mathbf{v_1}\) and \(\mathbf{v_2}\text{.}\)
This case produces a proper nodal sink. In the picture below, \(\lambda_1 = \lambda_2 < 0\text{.}\) The picture displays a vector field for a system with a single negative eigenvalue that corresponds to two linearly independent eigenvectors. Several solution curves are shown. The solution curves are lines flowing into the origin.
Vector field for a system with one negative eigenvalue of multiplicity 2
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{pt}(\mathbf{a}
\cos(qt)-\mathbf{b}\sin(qt)) +c_2e^{pt}(\mathbf{a}\sin(qt)+
\mathbf{b}\cos(qt))
\end{gather*}
.
Case 1: \(A\) has purely imaginary eigenvalues (\(\lambda_{1,2} = \pm qi\)) two linearly independent eigenvectors are \(\mathbf{v_{1,2}}=\mathbf{a}\pm\mathbf{b}i\text{.}\)
This case produces a center. In the picture below, \(\lambda_{1,2} = \pm qi\text{.}\) The picture displays a vector field for a system with purely imaginary eigenvalues. Several solution curves are shown. The solution curves are ellipses centered at the origin.
Vector field for a system with purely imaginary eigenvalues
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{pt}(\mathbf{a}
\cos(qt)-\mathbf{b}\sin(qt)) +c_2e^{pt}(\mathbf{a}\sin(qt)+
\mathbf{b}\cos(qt))\text{,}
\end{gather*}
continued.
Case 2: \(A\) has complex eigenvalues (\(\lambda_{1,2} = p \pm qi\)) with a positive real part (\(p >0 \)) and two linearly independent eigenvectors are \(\mathbf{v_{1,2}}=\mathbf{a}\pm\mathbf{b}i\text{.}\)
This case produces a spiral source. In the picture below, \(\lambda_{1,2} = p \pm qi\) with \(p>0\text{.}\) The picture displays a vector field for a system with complex eigenvalues that have a positive real part. Several solution curves are shown. The solution curves are spirals emanating from the origin.
Vector field for a system with complex eigenvalues that have a positive real part
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{pt}(\mathbf{a}
\cos(qt)-\mathbf{b}\sin(qt)) +c_2e^{pt}(\mathbf{a}\sin(qt)+
\mathbf{b}\cos(qt))\text{,}
\end{gather*}
continued.
Case 3: \(A\) has complex eigenvalues (\(\lambda_{1,2} = p \pm qi\)) with a negative real part (\(p < 0 \)) and two linearly independent eigenvectors are \(\mathbf{v_{1,2}}=\mathbf{a}\pm\mathbf{b}i\text{.}\)
This case produces a spiral sink. In the picture below, \(\lambda_{1,2} = p \pm qi\) with \(p < 0\text{.}\) The picture displays a vector field for a system with complex eigenvalues that have a negative real part. Several solution curves are shown. The solution curves are spirals flowing into the origin.
Vector field for a system with complex eigenvalues that have a negative real part
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{\lambda t}
\mathbf{v_1}+c_2e^{\lambda t}(t\mathbf{v_1}+\mathbf{v_2})
\end{gather*}
.
When the solution curves have the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{\lambda t}
\mathbf{v_1}+c_2e^{\lambda t}(t\mathbf{v_1}+\mathbf{v_2})\text{,}
\end{gather*}
the eigenvalue is real, has multiplicity 2, and there is only one linearly independent eigenvector corresponding to this eigenvalue.
Case 1: \(A\) has a single positive eigenvalue (\(\lambda_{1,2} = \lambda > 0\)) and only one linearly independent eigenvector. In other words, \(\lambda\) is a defective eigenvalue.
This case produces an improper nodal source. In the picture below, \(\lambda_{1,2} = \lambda >0\text{.}\) The picture displays a vector field for a system with a positive, defective eigenvalue. The solutions curve away from the origin.
Vector field for a system with a positive, defective eigenvalue
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{\lambda t}
\mathbf{v_1}+c_2e^{\lambda t}(t\mathbf{v_1}+\mathbf{v_2})\text{,}
\end{gather*}
continued.
Case 2: \(A\) has a single negative eigenvalue (\(\lambda_{1,2} = \lambda < 0\)) and only one linearly independent eigenvector. In other words, \(\lambda\) is a defective eigenvalue.
This case produces an improper nodal sink. In the picture below, \(\lambda_{1,2} = \lambda < 0\text{.}\) The picture displays a vector field for a system with a negative, defective eigenvalue. The solutions curve in toward the origin.
Vector field for a system with a negative, defective eigenvalue
Solution curves of the form
\begin{gather*}
\mathbf{x}(t)=c_1e^{\lambda t}
\mathbf{v_1}+c_2e^{\lambda t}(t\mathbf{v_1}+\mathbf{v_2})\text{,}
\end{gather*}
continued.
Case 3: \(A\) has a single eigenvalue that equals zero (\(\lambda_{1,2} = \lambda = 0\)) and only one linearly independent eigenvector. In other words, zero is defective eigenvalue.
This case produces parallel lines. In the picture below, \(\lambda_{1,2} = \lambda =0\text{.}\) The picture displays a vector field for a system with a zero, defective eigenvalue. It also displays the eigenvector. The solutions are lines that are parallel to the eigenvector. Half of the solutions flow in the direction of the eigenvector and the other half flow in the opposite direction.
Vector field for a system with a zero, defective eigenvalue
Summary.
| Eigenvalues | Origin and Phase Portrait |
|---|---|
| Two real | |
| \(\bullet\) opposite signs | \(\bullet\) saddle point |
| \(\bullet\) positive | \(\bullet\) improper nodal source |
| \(\bullet\) negative | \(\bullet\) improper nodal sink |
| Complex conjugate pair (\(p \pm qi\)) | |
| \(\bullet\) purely imaginary (\(p=0\)) | \(\bullet\) Center/Ellipses |
| \(\bullet\) \(p\) is positive | \(\bullet\) Spirial Source |
| \(\bullet\) \(p\) is negative | \(\bullet\) Spirial Sink |
| One nonzero real, \(\lambda \neq 0\) | |
| \(\bullet\) 2 linearly independent eigenvectors | \(\bullet\) Stars |
| \(\circ\) proper nodal source (\(\lambda >0\)) | |
| \(\circ\) proper nodal sink (\(\lambda <0\)) | |
| \(\bullet\) 1 linearly independent eigenvector | \(\bullet\) Improper nodal source or sink |
| \(\circ\) improper nodal source (\(\lambda >0\)) | |
| \(\circ\) improper nodal sink (\(\lambda <0\)) | |
| \(\lambda = 0\) | \(\bullet\) Parallel lines (\(\lambda_1=\lambda_2=0 \text{,}\) defective) |
| \(\bullet\) Parallel “rays” (\(0=\lambda_1 \neq \lambda_2)\) |
