Differential Equations: How to categorize graph and clockwise vs. counter-clockwise from eigenvalues?
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I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (CCW).
If someone could please tell me whether my understanding (typed below) is correct, I would really appreciate it!
Case 1: If $lambda_1,lambda_2>0$ and real, then it is a Source.
Case 2: If $lambda_1,lambda_2>0$ and complex, then it is a Spiral Source.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 3: If $lambda_1,lambda_2<0$ and real, then it is a Sink.
Case 4: If $lambda_1,lambda_2<0$ and complex, then it is a Spiral Sink.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 5: If $lambda_1=lambda_2=0$, then it is a Center.
Case 6: If $lambda_1>0>lambda_2$, then it is a Saddle.
Thank you in advance to anyone who can help!
ordinary-differential-equations complex-numbers eigenvalues-eigenvectors differential
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
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add a comment |
$begingroup$
I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (CCW).
If someone could please tell me whether my understanding (typed below) is correct, I would really appreciate it!
Case 1: If $lambda_1,lambda_2>0$ and real, then it is a Source.
Case 2: If $lambda_1,lambda_2>0$ and complex, then it is a Spiral Source.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 3: If $lambda_1,lambda_2<0$ and real, then it is a Sink.
Case 4: If $lambda_1,lambda_2<0$ and complex, then it is a Spiral Sink.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 5: If $lambda_1=lambda_2=0$, then it is a Center.
Case 6: If $lambda_1>0>lambda_2$, then it is a Saddle.
Thank you in advance to anyone who can help!
ordinary-differential-equations complex-numbers eigenvalues-eigenvectors differential
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
$endgroup$
add a comment |
$begingroup$
I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (CCW).
If someone could please tell me whether my understanding (typed below) is correct, I would really appreciate it!
Case 1: If $lambda_1,lambda_2>0$ and real, then it is a Source.
Case 2: If $lambda_1,lambda_2>0$ and complex, then it is a Spiral Source.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 3: If $lambda_1,lambda_2<0$ and real, then it is a Sink.
Case 4: If $lambda_1,lambda_2<0$ and complex, then it is a Spiral Sink.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 5: If $lambda_1=lambda_2=0$, then it is a Center.
Case 6: If $lambda_1>0>lambda_2$, then it is a Saddle.
Thank you in advance to anyone who can help!
ordinary-differential-equations complex-numbers eigenvalues-eigenvectors differential
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
$endgroup$
I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (CCW).
If someone could please tell me whether my understanding (typed below) is correct, I would really appreciate it!
Case 1: If $lambda_1,lambda_2>0$ and real, then it is a Source.
Case 2: If $lambda_1,lambda_2>0$ and complex, then it is a Spiral Source.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 3: If $lambda_1,lambda_2<0$ and real, then it is a Sink.
Case 4: If $lambda_1,lambda_2<0$ and complex, then it is a Spiral Sink.
Subcase 1: If $lambda_{re}>0$, then motion is CW.
Subcase 2: If $lambda_{re}<0$, then motion is CCW.
Case 5: If $lambda_1=lambda_2=0$, then it is a Center.
Case 6: If $lambda_1>0>lambda_2$, then it is a Saddle.
Thank you in advance to anyone who can help!
ordinary-differential-equations complex-numbers eigenvalues-eigenvectors differential
ordinary-differential-equations complex-numbers eigenvalues-eigenvectors differential
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
asked Dec 11 '18 at 23:41
Mathematic314Mathematic314
83
83
add a comment |
add a comment |
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