(M, H)
[[M == 0, H == 0], [M == 18, H == 0], [M == 0, H == -20], [M == 12, H == 4]]
[ 0.360000000000000 0]
[ 0 -0.100000000000000]
[-0.360000000000000 -0.540000000000000]
[ 0 0.0800000000000000]
[ 0.960000000000000 0]
[-0.200000000000000 0.100000000000000]
[ -0.240000000000000 -0.360000000000000]
[ 0.0400000000000000 -0.0200000000000000]
(M, H)
(N, P)
[[N == 0, P == 0], [N == -1000, P == 0], [N == 3000, P == 0], [N == -500/7*sqrt(45109) - 105500/7, P == -50/49*sqrt(45109) - 10550/49], [N == 500/7*sqrt(45109) - 105500/7, P == 50/49*sqrt(45109) - 10550/49]]
[1/1400*(1185703*sqrt(45109) - sqrt(-601256621013526*sqrt(45109) + 127700168580419822) - 251829341)/(197*sqrt(45109) - 41959), 1/1400*(1185703*sqrt(45109) + sqrt(-601256621013526*sqrt(45109) + 127700168580419822) - 251829341)/(197*sqrt(45109) - 41959)]
(N, P)
[[N == 0, P == 0], [N == -1000, P == 0], [N == 3000, P == 0], [N == -500/7*sqrt(613) + 2500/7, P == -50/49*sqrt(613) + 250/49], [N == 500/7*sqrt(613) + 2500/7, P == 50/49*sqrt(613) + 250/49]]
'A change in w resulted in a change in stability'
[-1/1400*(3679*sqrt(613) + sqrt(134733626*sqrt(613) + 4000114862) + 65827)/(19*sqrt(613) + 487), -1/1400*(3679*sqrt(613) - sqrt(134733626*sqrt(613) + 4000114862) + 65827)/(19*sqrt(613) + 487)]
'The arrows are pointing the opposite way. That change in stability is the result of a Hopf Birurcation'
'The simplified equation is 0.0008*G*(1+0.008*G^12)-1=0'
G
2.508750670737234
0.501750134147447
0.100350026829489
'It is an unstabe eq point'
Error in lines 1-1
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
flags=compile_flags) in namespace, locals
File "", line 1, in <module>
TypeError: unsupported operand type(s) for ** or pow(): 'sage.symbolic.expression.Expression' and 'function'
G
5.516960270495793
1.10339205409916
0.220678410819832
'It is a stable spiral eq point'
Interact: please open in CoCalc
'The birfurcation ocuurs when n equals about 7.5'
'It is much easier to find when a Hopf birurcation occurs on Sage because it is less time consuming than doing it by hand and it does not leave much room for mistakes or inconsistencies'
'The eq point should undergo a change in stability in order for a Hopf Bifurcation to occcur. Detecting it numerically, although very time consuming, can be accomplished by seeing how a change in parameter values can affect the (negative or positive) signs of the eigenvalues. If an eigenvalue changes from being negative to positive or vice versa, a Hopf bifucation has occured'
Error in lines 1-12
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute
flags=compile_flags) in namespace, locals
File "<string>", line 4
Hprime=
^
SyntaxError: invalid syntax