General Phase Portrait (v2)

Expression 49: "x" Subscript, 1 "r" "e" , Baseline left parenthesis, "t" , right parenthesis equals "e" Superscript, lambda Subscript, "r" "e" , Baseline "t" , Baseline cosine left parenthesis, lambda Subscript, "i" "m" , Baseline "t" , right parenthesis

NOTE: technically this component can be imaginary (scaled by i), but because of the principle of superposition (solutions to linear combinations of solutions to linear homogeneous systems form another solution) we can actually just remove the imaginary scaling

Expression 51: "x" Subscript, 1 "i" "m" , Baseline left parenthesis, "t" , right parenthesis equals "e" Superscript, lambda Subscript, "r" "e" , Baseline "t" , Baseline sine left parenthesis, lambda Subscript, "i" "m" , Baseline "t" , right parenthesis

Expression 52: "x" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis equals "c" Subscript, 1 , Baseline "x" Subscript, 1 "r" "e" , Baseline left parenthesis, "t" , right parenthesis plus "c" Subscript, 2 , Baseline "x" Subscript, 1 "i" "m" , Baseline left parenthesis, "t" , right parenthesis

Expression 53: "x" Subscript, 2 "r" "e" , Baseline left parenthesis, "t" , right parenthesis equals "f" Subscript, "c" "o" "m" "p" "M" "u" "l" "t" "R" "e" , Baseline left parenthesis, "x" Subscript, 1 "r" "e" , Baseline left parenthesis, "t" , right parenthesis , "x" Subscript, 1 "i" "m" , Baseline left parenthesis, "t" , right parenthesis , "v" Subscript, 2 "r" "e" , Baseline , "v" Subscript, 2 "i" "m" , Baseline , right parenthesis

Expression 54: "x" Subscript, 2 "i" "m" , Baseline left parenthesis, "t" , right parenthesis equals "f" Subscript, "c" "o" "m" "p" "M" "u" "l" "t" "I" "m" , Baseline left parenthesis, "x" Subscript, 1 "r" "e" , Baseline left parenthesis, "t" , right parenthesis , "x" Subscript, 1 "i" "m" , Baseline left parenthesis, "t" , right parenthesis , "v" Subscript, 2 "r" "e" , Baseline , "v" Subscript, 2 "i" "m" , Baseline , right parenthesis

Expression 55: "x" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis equals "c" Subscript, 1 , Baseline "x" Subscript, 2 "r" "e" , Baseline left parenthesis, "t" , right parenthesis plus "c" Subscript, 2 , Baseline "x" Subscript, 2 "i" "m" , Baseline left parenthesis, "t" , right parenthesis

Expression 56: left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , "x" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis

domain t Minimum: negative 100

less than or equal to "t" less than or equal to

domain t Maximum: 100

solution with real eigenvalues

Expression 58: "X" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis equals "c" Subscript, 1 , Baseline "e" Superscript, lambda Subscript, 1 , Baseline "t" , Baseline "v" Subscript, 1 , Baseline plus "c" Subscript, 2 , Baseline "e" Superscript, lambda Subscript, 2 , Baseline "t" , Baseline "u" Subscript, 1 , Baseline

Expression 59: "X" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis equals "c" Subscript, 1 , Baseline "e" Superscript, lambda Subscript, 1 , Baseline "t" , Baseline "v" Subscript, 2 , Baseline plus "c" Subscript, 2 , Baseline "e" Superscript, lambda Subscript, 2 , Baseline "t" , Baseline "u" Subscript, 2 , Baseline

Expression 60: left parenthesis, "X" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , "X" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis

domain t Minimum: negative 10

less than or equal to "t" less than or equal to

domain t Maximum: 10

Expression 61: left brace, "k" less than 0 : left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , "x" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis , left parenthesis, "X" Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , "X" Subscript, 2 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis , right brace

domain t Minimum: negative 10

less than or equal to "t" less than or equal to

domain t Maximum: 10

Hidden Label: left brace, "k" less than 0 : left parenthesis, "x" Subscript, 1 , Baseline left parenthesis, 0 , right parenthesis , "x" Subscript, 2 , Baseline left parenthesis, 0 , right parenthesis , right parenthesis , left parenthesis, "X" Subscript, 1 , Baseline left parenthesis, 0 , right parenthesis , "X" Subscript, 2 , Baseline left parenthesis, 0 , right parenthesis , right parenthesis , right brace

Label

equals

left parenthesis, negative 0.6 2 5 3 , negative 0.9 3 6 3 , right parenthesis

left parenthesis, negative 1.3 3 9 , 0.1 7 6 5 , right parenthesis

left parenthesis, negative 1.2 6 8 , 1.1 8 6 , right parenthesis

left parenthesis, negative 0.4 5 4 8 , 1.5 0 1 , right parenthesis

left parenthesis, 0.6 2 5 3 , 0.9 3 6 3 , right parenthesis

left parenthesis, 1.3 3 9 , negative 0.1 7 6 5 , right parenthesis

left parenthesis, 1.2 6 8 , negative 1.1 8 6 , right parenthesis

left parenthesis, 0.4 5 4 8 , negative 1.5 0 1 , right parenthesis

Initial Conditions

Expression 64: "n" equals 8

Expression 65: "P" equals StartFraction, 2 pi Over "n" , EndFraction times left bracket, 1... "n" , right bracket

equals

0.7 8 5 3 9 8 1 6 3 3 9 7

1.5 7 0 7 9 6 3 2 6 7 9

2.3 5 6 1 9 4 4 9 0 1 9

3.1 4 1 5 9 2 6 5 3 5 9

3.9 2 6 9 9 0 8 1 6 9 9

4.7 1 2 3 8 8 9 8 0 3 8

5.4 9 7 7 8 7 1 4 3 7 8

6.2 8 3 1 8 5 3 0 7 1 8

Expression 66: "c" Subscript, 1 , Baseline equals cosine left parenthesis, "P" plus "T" , right parenthesis

equals

0.3 2 1 5 8 0 4 8 6 7 6 2

negative 0.4 4 2 1 5 5 1 1 5 2 6 9

negative 0.9 4 6 8 8 2 2 4 7 4 4 9

negative 0.8 9 6 9 3 8 6 0 1 0 4 3

negative 0.3 2 1 5 8 0 4 8 6 7 6 2

0.4 4 2 1 5 5 1 1 5 2 6 9

0.9 4 6 8 8 2 2 4 7 4 4 9

0.8 9 6 9 3 8 6 0 1 0 4 3

Expression 67: "c" Subscript, 2 , Baseline equals negative sine left parenthesis, "P" plus "T" , right parenthesis

equals

negative 0.9 4 6 8 8 2 2 4 7 4 4 9

negative 0.8 9 6 9 3 8 6 0 1 0 4 3

negative 0.3 2 1 5 8 0 4 8 6 7 6 2

0.4 4 2 1 5 5 1 1 5 2 6 9

0.9 4 6 8 8 2 2 4 7 4 4 9

0.8 9 6 9 3 8 6 0 1 0 4 3

0.3 2 1 5 8 0 4 8 6 7 6 2

negative 0.4 4 2 1 5 5 1 1 5 2 6 9

Hidden Label: left parenthesis, "c" Subscript, 1 , Baseline , "c" Subscript, 2 , Baseline , right parenthesis

Label

equals

left parenthesis, 0.3 2 1 6 , negative 0.9 4 6 9 , right parenthesis

left parenthesis, negative 0.4 4 2 2 , negative 0.8 9 6 9 , right parenthesis

left parenthesis, negative 0.9 4 6 9 , negative 0.3 2 1 6 , right parenthesis

left parenthesis, negative 0.8 9 6 9 , 0.4 4 2 2 , right parenthesis

left parenthesis, negative 0.3 2 1 6 , 0.9 4 6 9 , right parenthesis

left parenthesis, 0.4 4 2 2 , 0.8 9 6 9 , right parenthesis

left parenthesis, 0.9 4 6 9 , 0.3 2 1 6 , right parenthesis

left parenthesis, 0.8 9 6 9 , negative 0.4 4 2 2 , right parenthesis

"C" Subscript, 1 , Baseline

"C" Subscript, 2 , Baseline