It is apparent that certain chaotic systems can be written in a rather compact fashion using complex notation. For example AC6 and AC7 from Sprott (page 157):
where z = a + ib.
Since the imaginary part is just that - imaginary - I cannot deal with it directly on my (real) computer. So after a little algebra we re-cast from the compact to the less than compact pair of equations:
where, in the second equation, the plus sign corresponds to AC6, and the minus sign to AC7. At least now we can imagine that the imaginary part is real (i.e. b).
Interestingly, this is a real mess to patch together (in the real world) - perhaps this says something about the universe...
Incidentally, this is the first time I've used the computer's power functions (to get the cubes) - via AD538s.
Initial conditions are a = 0.5, b = 0.5; their first derivatives are zero. And my state space plots:
Both plots: b is vertical and a horizontal (equal scales throughout: 0.2 V/cm.)
These compare nicely with Sprott's Fig 6.28.
Since the imaginary part is just that - imaginary - I cannot deal with it directly on my (real) computer. So after a little algebra we re-cast from the compact to the less than compact pair of equations:
Interestingly, this is a real mess to patch together (in the real world) - perhaps this says something about the universe...
Incidentally, this is the first time I've used the computer's power functions (to get the cubes) - via AD538s.
Initial conditions are a = 0.5, b = 0.5; their first derivatives are zero. And my state space plots:
 |
| AC6 |
 |
| AC7 |
Both plots: b is vertical and a horizontal (equal scales throughout: 0.2 V/cm.)
These compare nicely with Sprott's Fig 6.28.
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