-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathBisection_1_29.m
597 lines (507 loc) · 22 KB
/
Bisection_1_29.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
% addpath(genpath('/home/zhw22003/PRE-Model/YALMIP-master'))
% system('export PATH=$PATH:/home/zhw22003/PRE-Model/mosek/10.2/tools/platform/linux64x86/bin')
% addpath(genpath('/home/zhw22003/PRE-Model/mosek'))
Re_list = logspace(log10(200), log10(2000), 16);
delta_list = logspace(-6, 0, 200);
% Re_list = logspace(log10(200), log10(2000), 4);
% delta_list = logspace(-6, 0, 2);
% Re_list= Re_list(1:5);
% delta_list=delta_list(1:2);
% epsilon = 0.01;
B = eye(9);
C = eye(9);
D = zeros(9,9);
n = [1;0;0;0;0;0;0;0;-1];
% gamma_results = zeros(length(Re_list), length(delta_list));
% test_u_delta = zeros(length(Re_list), length(delta_list));
% test_u_d_gamma = zeros(length(Re_list), length(delta_list));
% reachable_delta = zeros(length(Re_list), length(delta_list));
% lambda_min_results = zeros(length(Re_list), length(delta_list));
Lx = 1.75*pi;
Lz = 1.2*pi;
alpha = (2*pi)/Lx;
Beta = pi/2;
Gamma = 2*pi/Lz;
KBG = sqrt(Beta^2+Gamma^2);
KAG = sqrt(alpha^2+Gamma^2);
KABG = sqrt(alpha^2+Beta^2+Gamma^2);
[RHS_J_mean_shear, nonlinear, u] = nonliner(alpha,Beta,Gamma,KBG, KAG,KABG);
delete(gcp('nocreate'));
% parpool(32);
forcing_bi = zeros(length(Re_list), length(delta_list));
for ind_Re = 1:length(Re_list)
Re = Re_list(ind_Re);
[local_forcing_amp]=LMI_Re(Re,delta_list,RHS_J_mean_shear, nonlinear,u,B,C,alpha,Beta,Gamma,KBG, KAG,KABG);
forcing_bi(ind_Re,:) = local_forcing_amp;
end
save Bisection_2_5.mat
function [local_forcing_amp, local_u_bisection, local_mid_u_bisection]=LMI_Re(Re,delta_list,RHS_J_mean_shear, nonlinear, u,B,C, alpha,Beta,Gamma,KBG, KAG,KABG)
local_forcing_amp = NaN*ones(1, length(delta_list)); % 1 x length(delta_list)
local_u_bisection = cell(1, length(delta_list)); % Cell array of size 1 x length(delta_list)
local_mid_u_bisection = NaN*ones(1, length(delta_list)); % 1 x length(delta_list)
% Initialize other variables
local_u_upper_bound = NaN*ones(1, length(delta_list));
local_deltaf = NaN*ones(1, length(delta_list));
linear_term = [(Beta^2)/Re;
((4*Beta^2)/3 + Gamma^2)/Re;
(Beta^2+Gamma^2)/Re;
(3*alpha^2+4*Beta^2)/(3*Re);
(alpha^2+Beta^2)/Re;
(3*alpha^2+4*Beta^2+3*Gamma^2)/(3*Re);
(alpha^2+Beta^2+Gamma^2)/Re;
(alpha^2+Beta^2+Gamma^2)/Re;
(9*Beta^2)/Re];
RHS_R_viscous = diag(double(linear_term));
A = RHS_J_mean_shear - RHS_R_viscous;
for ind_delta = 1:length(delta_list)
yalmip('clear');
delta = delta_list(ind_delta);
delta2 = delta^2;
% if ~isempty(n) %we have a non-zero null space of nonlinear term.
% n = double(n);
% lambda=sdpvar(length(A),1);
% kappa=sdpvar(length(A),1);
% %n=double(n);
% %implementing the inequality constraint like the Lagrange
% %multiplier
% for ind_e=1:length(A)
% e=zeros(length(A),1);
% e(ind_e)=1;
% lambda_af=lambda_af+lambda(ind_e,1)*e*n';
% kappa_ff=kappa_ff+kappa(ind_e,1)*(e*n'+n*e');
% end
% else
% lambda=sdpvar(1,1);
% lambda_af=zeros(size(A));
% kappa=sdpvar(1,1);
% kappa_ff=zeros(size(A));
% end
[lambda, kappa, lambda_af, kappa_ff] = check(A, C, B);
[F_square] = F__square(nonlinear, u);
% s = sdpvar(length(F_square), 1);
% s_bound = zeros(size(A));
% for m_ind = 1:length(F_square)
% s_bound = s_bound + s(m_ind) * delta2 * double(F_square{m_ind});
% end
%
% if length(s) <= length(A)
% diag_s = diag(s);
% else
% diag_s = diag(s(1:length(A))) + eye(size(A)) * s(length(A) + 1);
% end
[s, diag_s, s_bound] = s_diag_s(F_square, A, delta2);
% Construct LMI conditions
epsilon=0.01;
sdp_options = sdpsettings('solver', 'mosek');
options.bisection.absgaptol = 1e-16;
[result_yalmip, P, lambda_af, kappa_ff] = LMI(A,s_bound, epsilon, lambda_af, kappa_ff, diag_s, B, C, s, sdp_options);
if result_yalmip.problem == 0
lambda_af_value = value(lambda_af); % Lagrange mutiplier
kappa_ff_value = value(kappa_ff); % Lagrange mutiplier
optp = value(P);
umax = max(eig(optp)); % c2
umin = min(eig(optp)); % c1
c3 = epsilon;
c2 = umax;
c1 = umin;
L = norm(B);
c4 = norm(2*value(P));
local_deltaf(ind_delta) = sqrt(value(delta2)) * sqrt(umin / umax);
eta_1 = norm(C);
eta_2 = 0;
local_u_upper_bound(ind_delta) = (c1*c3*delta)/(c2*c4*L);
disp(['Feasible solution found for Re = ', num2str(Re), ', delta = ', num2str(delta_list(ind_delta))]);
disp(['P: ', mat2str(optp)]);
disp(['umax: ', num2str(umax)]);
disp(['umin: ', num2str(umin)]);
T = 20000;
omega=1;
[local_forcing_amp, local_u_bisection, local_mid_u_bisection] = compute_norm_u(Re, A, B, T, omega,local_deltaf(ind_delta), local_u_upper_bound(ind_delta), delta_list, alpha,Beta,Gamma,KBG, KAG,KABG);
else
disp(['No feasible solution found for Re = ', num2str(Re), ', delta = ', num2str(delta_list(ind_delta))]);
disp(['YALMIP error: ', yalmiperror(result_yalmip.problem)]);
end
end
end
function [RHS_J_mean_shear,nonlinear, u] = nonliner(alpha,Beta,Gamma,KBG, KAG,KABG)
u = sym('u', [9, 1]);
term1 = -sqrt(3/2)*Beta*Gamma*u(6)*u(8)/KABG+sqrt(3/2)*Beta*Gamma*u(2)*u(3)/KBG;
term2 = (5/3)*sqrt(2/3)*Gamma^2*u(4)*u(6)/KAG-Gamma^2*u(5)*u(7)/(sqrt(6)*KAG) ...
-alpha*Beta*Gamma*u(5)*u(8)/(sqrt(6)*KAG*KABG)-sqrt(3/2)*Beta*Gamma*u(1)*u(3)/KBG-sqrt(3/2)*Beta*Gamma*u(3)*u(9)/KBG;
term3 = 2*alpha*Beta*Gamma*(u(4)*u(7)+u(5)*u(6))/(sqrt(6)*KAG*KBG)+(Beta^2*(3*alpha^2+Gamma^2)-3*Gamma^2*(alpha^2+Gamma^2))*u(4)*u(8)/(sqrt(6)*KAG*KBG*KABG);
term4 = -alpha*u(1)*u(5)/sqrt(6)-10*alpha^2*u(2)*u(6)/(3*sqrt(6)*KAG) ...
-sqrt(3/2)*alpha*Beta*Gamma*u(3)*u(7)/KAG*KBG-sqrt(3/2)*alpha^2*Beta^2*u(3)*u(8)/KAG*KBG*KABG-alpha*u(5)*u(9)/sqrt(6);
term5 = alpha*u(1)*u(4)/sqrt(6)+alpha^2*u(2)*u(7)/(sqrt(6)*KAG)-alpha*Beta*Gamma*u(2)*u(8)/(sqrt(6)*KAG*KABG)+alpha*u(4)*u(9)/sqrt(6)+2*alpha*Beta*Gamma*u(3)*u(6)/(sqrt(6)*KAG*KBG);
term6 = alpha*u(1)*u(7)/sqrt(6)+sqrt(3/2)*Beta*Gamma*u(1)*u(8)/KABG ...
+10*(alpha^2-Gamma^2)*u(2)*u(4)/(KAG*3*sqrt(6))-2*sqrt(2/3)*u(3)*u(5)*alpha*Beta*Gamma/(KAG*KBG)+alpha*u(7)*u(9)/sqrt(6)+sqrt(3/2)*Beta*Gamma*u(8)*u(9)/KABG;
term7 = -alpha*(u(1)*u(6)+u(6)*u(9))/sqrt(6)+(Gamma^2-alpha^2)*u(2)*u(5)/(sqrt(6)*KAG)+alpha*Beta*Gamma*u(3)*u(4)/(sqrt(6)*KAG*KBG);
term8 = 2*alpha*Beta*Gamma*u(2)*u(5)/(sqrt(6)*KAG*KABG)+Gamma^2*(3*alpha^2-Beta^2+3*Gamma^2)*u(3)*u(4)/(sqrt(6)*KAG*KBG*KABG);
term9 = sqrt(3/2)*Beta*Gamma*u(2)*u(3)/KBG-sqrt(3/2)*Beta*Gamma*u(6)*u(8)/KABG;
nonlinear= [term1;
term2;
term3;
term4;
term5;
term6;
term7;
term8;
term9];
nonlinear_gradient = sym(zeros(length(u), length(u)));
for i = 1:length(nonlinear)
nonlinear_gradient(i, :) = gradient(nonlinear(i), u);
end
%nonlinear_gradinet = gradient(nonlinear, u);
a_bar = [1;0;0;0;0;0;0;0;0];
nonlinear_gradient_sub = double(subs(nonlinear_gradient, u, a_bar));
RHS_J_mean_shear = nonlinear_gradient_sub;
end
function [F_square] = F__square(nonlinear, u)
F_square = cell(1, length(nonlinear));
for n_ind = 1:length(nonlinear)
%initialize F{n_ind}
F{n_ind} = sym(zeros(length(u), length(u)));
for x_ind = 1:length(u)
for y_ind = 1:length(u)
F{n_ind}(x_ind, y_ind) = 1/2 * diff(diff(nonlinear(n_ind), u(x_ind)), u(y_ind));
end
end
%convert F{n_ind} to double if it's symbolic
[V, D] = eig(double(F{n_ind}));
%store F_square as double
F_square{n_ind} = double(V * D^2 /(V));
end
end
%% LMI
function[result_yalmip, P,lambda_af,kappa_ff] = LMI(A,s_bound, epsilon, lambda_af, kappa_ff, diag_s, B, C, s, sdp_options)
P = sdpvar(size(A, 1), size(A, 2));
G11 = A' * P + P * A + s_bound + epsilon*eye(size(A));
G12 = P*B + C'*lambda_af;
G22 = -diag_s + kappa_ff;
% LMI
G = [G11, G12;
G12', G22;];
V_ineq = P - epsilon * eye(size(A));
% Solve the LMI
F = [V_ineq >= 0, G <= 0, s >= 0];
result_yalmip = optimize(F, [], sdp_options);
end
function[s, diag_s, s_bound] = s_diag_s(F_square, A, delta2)
s = sdpvar(length(F_square), 1);
s_bound = zeros(size(A));
for m_ind = 1:length(F_square)
s_bound = s_bound + s(m_ind) * delta2 * double(F_square{m_ind});
end
if length(s) <= length(A)
diag_s = diag(s);
else
diag_s = diag(s(1:length(A))) + eye(size(A)) * s(length(A) + 1);
end
end
function[lambda, kappa, lambda_af, kappa_ff] = check(A, C, B)
n = [1;0;0;0;0;0;0;0;-1];
lambda_af=zeros(size(C,1),size(C,1));
kappa_ff=zeros(size(B,2),size(B,2));
if ~isempty(n) %%we have a non-zero null space of nonlinear term.
n = double(n);
lambda=sdpvar(length(A),1);
kappa=sdpvar(length(A),1);
%n=double(n);
%implementing the inequality constraint like the Lagrange
%multiplier
for ind_e=1:length(A)
e=zeros(length(A),1);
e(ind_e)=1;
lambda_af=lambda_af+lambda(ind_e,1)*e*n';
kappa_ff=kappa_ff+kappa(ind_e,1)*(e*n'+n*e');
end
else
lambda=sdpvar(1,1);
lambda_af=zeros(size(A));
kappa=sdpvar(1,1);
kappa_ff=zeros(size(A));
end
end
function [local_forcing_amp, local_u_bisection, local_mid_u_bisection] = compute_norm_u(Re, A, B, T, omega, deltaf, u_upper_bound, delta_list, alpha,Beta,Gamma,KBG, KAG,KABG)
ini=randn(9,1);
u0 = deltaf*ini/norm(ini);
% u0 =zeros(9,1);
T = 2000;
tspan = linspace(0, T, 10000);
I = eye(size(A));
C = eye(9);
D = zeros(9, 9);
G = C * inv(1i * omega * I - A) * B;
max_SVD = max(svd(G));
sys = ss(A, B, C, D);
% u0 = zeros(9,1);
% tol = 1e-9; % Tolerance for convergence
% max_iter = 1000; % Maximum fixed-point iterations per step
% dt = diff(tspan);
% dt = dt(1);
%[t, u, forcing] = ode45(@(t,u) ode_system_123(t, u, A, B, u_upper_bound), tspan, u0);
[t, u, forcing] = rk5_solver(@(t,u) ode_system_123(t, u, A, B, u_upper_bound, alpha,Beta,Gamma,KBG, KAG,KABG), tspan, u0);
% Linear Equation analytical solution
% for t_ind = 1:length(tspan)
% syms t_sym real; % Create a symbolic variable for time
% u_sym = expm(A *tspan(t_ind)) * u0; % Analytical solution: u(t)=exp(At)*u0
% u_analytical(:,t_ind) = double(u_sym); % Store symbolic expression
% end
%old norm integrate over time
dt=diff(tspan);
dt=dt(1);
% u_square=sum(u.^2,2);
% norm_u=sqrt(sum(u_square(2:end).*dt)/max(tspan));
%1 norm maximal over time
a_square=sqrt(sum(u.^2,2));
%norm_u = sqrt(sum(u_square.^2)*dt);
y =C*u';
y = y'; % output
norm_u=max(a_square);
% %Bisection
max_u = 0.01;
min_u = 0;
% Tolerance and maximum iterations
tol = 1e-14;
maxIter = 100;
% Call the bisection method
% u0=zeros(9,1);
% [forcing_amp, iterations] = bisectionMethod(min_u, max_u, tol, maxIter, A, B,tspan, u0);
num_simulations = 20; % Number of simulations per step
[local_forcing_amp, iterations, local_u_bisection, local_mid_u_bisection] = bisectionMethodMulti(min_u, max_u, tol, maxIter, A, B, tspan, u0, num_simulations,deltaf,delta_list,alpha,Beta,Gamma,KBG, KAG,KABG);
% Display results
fprintf('Approximated root: %.20f\n', local_forcing_amp);
fprintf('Number of iterations: %d\n', iterations);
%
% %% t vs local_mid_u_bisection
% t = linspace(0, 10, 40); % Creates a 1x40 time vector from 0 to 10
% loglog(t, local_mid_u_bisection);
% xlabel('Time');
% ylabel('Mid-u Values');
% title('Log-Log Plot of Mid-u vs Time');
%
% %% t vs local_u_bisection
% % Initialize a vector to store the maximum values
% max_values = zeros(40, 1); % 40 iterations
%
% % Loop through each iteration
% for iter = 1:40
% % Initialize a temporary vector to store max values for this iteration
% temp_max = zeros(20, 1); % 20 simulations per iteration
%
% % Loop through each simulation
% for sim = 1:20
% data = local_u_bisection{iter, sim}; % Extract data from the cell
%
% % Compute the maximum value for this simulation
% % Use max(data(:)) to ensure a scalar is returned
% temp_max(sim) = max(data(:));
% end
%
% % Store the maximum value across all simulations for this iteration
% max_values(iter) = max(temp_max);
% end
%
% % Create a time vector (iteration numbers)
% t = 1:40;
%
% % Plot the maximum values
% loglog(t, max_values, '-o'); % Plot with circles for each point
% xlabel('Iteration');
% ylabel('Maximum Value');
% title('Log-Log Plot of Maximum Values Across Simulations');
% grid on;
end
function [du_dt, forcing] = ode_system_123(t, u, A, B, u_upper_bound, alpha,Beta,Gamma,KBG, KAG,KABG)
force=randn(9,1);
forcing=u_upper_bound*force/norm(force);
term1 = -sqrt(3/2)*Beta*Gamma*u(6)*u(8)/KABG+sqrt(3/2)*Beta*Gamma*u(2)*u(3)/KBG;
term2 = (5/3)*sqrt(2/3)*Gamma^2*u(4)*u(6)/KAG-Gamma^2*u(5)*u(7)/(sqrt(6)*KAG) ...
-alpha*Beta*Gamma*u(5)*u(8)/(sqrt(6)*KAG*KABG)-sqrt(3/2)*Beta*Gamma*u(1)*u(3)/KBG-sqrt(3/2)*Beta*Gamma*u(3)*u(9)/KBG;
term3 = 2*alpha*Beta*Gamma*(u(4)*u(7)+u(5)*u(6))/(sqrt(6)*KAG*KBG)+(Beta^2*(3*alpha^2+Gamma^2)-3*Gamma^2*(alpha^2+Gamma^2))*u(4)*u(8)/(sqrt(6)*KAG*KBG*KABG);
term4 = -alpha*u(1)*u(5)/sqrt(6)-10*alpha^2*u(2)*u(6)/(3*sqrt(6)*KAG) ...
-sqrt(3/2)*alpha*Beta*Gamma*u(3)*u(7)/KAG*KBG-sqrt(3/2)*alpha^2*Beta^2*u(3)*u(8)/KAG*KBG*KABG-alpha*u(5)*u(9)/sqrt(6);
term5 = alpha*u(1)*u(4)/sqrt(6)+alpha^2*u(2)*u(7)/(sqrt(6)*KAG)-alpha*Beta*Gamma*u(2)*u(8)/(sqrt(6)*KAG*KABG)+alpha*u(4)*u(9)/sqrt(6)+2*alpha*Beta*Gamma*u(3)*u(6)/(sqrt(6)*KAG*KBG);
term6 = alpha*u(1)*u(7)/sqrt(6)+sqrt(3/2)*Beta*Gamma*u(1)*u(8)/KABG ...
+10*(alpha^2-Gamma^2)*u(2)*u(4)/(KAG*3*sqrt(6))-2*sqrt(2/3)*u(3)*u(5)*alpha*Beta*Gamma/(KAG*KBG)+alpha*u(7)*u(9)/sqrt(6)+sqrt(3/2)*Beta*Gamma*u(8)*u(9)/KABG;
term7 = -alpha*(u(1)*u(6)+u(6)*u(9))/sqrt(6)+(Gamma^2-alpha^2)*u(2)*u(5)/(sqrt(6)*KAG)+alpha*Beta*Gamma*u(3)*u(4)/(sqrt(6)*KAG*KBG);
term8 = 2*alpha*Beta*Gamma*u(2)*u(5)/(sqrt(6)*KAG*KABG)+Gamma^2*(3*alpha^2-Beta^2+3*Gamma^2)*u(3)*u(4)/(sqrt(6)*KAG*KBG*KABG);
term9 = sqrt(3/2)*Beta*Gamma*u(2)*u(3)/KBG-sqrt(3/2)*Beta*Gamma*u(6)*u(8)/KABG;
nonlinear= [term1;
term2;
term3;
term4;
term5;
term6;
term7;
term8;
term9];
du_dt = A * u + B * forcing+ nonlinear;
end
function [t, x, forcing] = rk5_solver(ode_func, tspan, u0)
% RK5 Solver for ODEs
% Inputs:
% ode_func - Function handle for the ODE system (f(t, u))
% tspan - Time vector [t0, t1, ..., tn]
% u0 - Initial condition (row vector)
% Outputs:
% t - Time vector (same as tspan)
% x - Solution matrix (size: length(tspan) x length(u0))
% forcing - Forcing term evaluated at each step (optional)
% Number of time steps
N = length(tspan);
% Preallocate solution and forcing arrays
x = zeros(N, length(u0));
forcing = zeros(N, length(u0));
x(1, :) = u0; % Set initial condition
% Coefficients for RK5 (Dormand-Prince example)
c = [0, 1/4, 3/8, 12/13, 1, 1/2];
b = [16/135, 0, 6656/12825, 28561/56430, -9/50, 2/55];
a = [
0, 0, 0, 0, 0, 0;
1/4, 0, 0, 0, 0, 0;
3/32, 9/32, 0, 0, 0, 0;
1932/2197, -7200/2197, 7296/2197, 0, 0, 0;
439/216, -8, 3680/513, -845/4104, 0, 0;
-8/27, 2, -3544/2565, 1859/4104, -11/40, 0
];
% Time-stepping loop
for i = 1:N-1
h = tspan(i+1) - tspan(i); % Time step size
u_current = x(i, :)'; % Current state
% Compute RK5 coefficients (k1 to k6)
k1 = h * ode_func(tspan(i), u_current);
k2 = h * ode_func(tspan(i) + c(2)*h, u_current + a(2,1)*k1);
k3 = h * ode_func(tspan(i) + c(3)*h, u_current + a(3,1)*k1 + a(3,2)*k2);
k4 = h * ode_func(tspan(i) + c(4)*h, u_current + a(4,1)*k1 + a(4,2)*k2 + a(4,3)*k3);
k5 = h * ode_func(tspan(i) + c(5)*h, u_current + a(5,1)*k1 + a(5,2)*k2 + a(5,3)*k3 + a(5,4)*k4);
k6 = h * ode_func(tspan(i) + c(6)*h, u_current + a(6,1)*k1 + a(6,2)*k2 + a(6,3)*k3 + a(6,4)*k4 + a(6,5)*k5);
% Update state
x(i+1, :) = (u_current + b(1)*k1 + b(2)*k2 + b(3)*k3 + b(4)*k4 + b(5)*k5 + b(6)*k6)';
% Optionally, store forcing term for analysis
[~, forcing(i+1, :)] = ode_func(tspan(i), u_current);
end
t = tspan; % Output time vector
end
function [local_forcing_amp, iterations, local_u_bisection, local_mid_u_bisection] = bisectionMethodMulti(min_u, max_u, tol, maxIter, A, B, tspan, u0, num_simulations, deltaf, delta_list, alpha,Beta,Gamma,KBG, KAG,KABG)
local_u_bisection = cell(maxIter, num_simulations); % Preallocate cell array
local_mid_u_bisection = NaN * ones(maxIter, 1); % Preallocate array for mid_u values
% Modified Bisection Method with multiple simulations per step
for iter = 1:maxIter
mid_u = (min_u + max_u) / 2;
norm_values = zeros(num_simulations, 1);
% Run multiple simulations
for sim = 1:num_simulations
ini = randn(9, 1);
u0_sim = deltaf * ini / norm(ini);
% u0_sim = zeros(9, 1); % Reset initial condition
[~, u, ~] = rk5_solver(@(t, u) ode_system_123(t, u, A, B, mid_u, alpha,Beta,Gamma,KBG, KAG,KABG), tspan, u0_sim);
% Store u for this simulation and iteration
local_u_bisection{iter, sim} = u;
% Calculate norm only for late time (last 50% of the time domain)
late_time_indices = round(0.9 * length(tspan)):length(tspan);
norm_values(sim) = mean(sqrt(sum(u(late_time_indices, :).^2, 2))); % Average norm over late time
% norm_values_timehistory = sqrt(sum(u(late_time_indices, :).^2, 2)); % t over norm_values_timehistory
end
% Store mid_u for this iteration
local_mid_u_bisection(iter) = mid_u;
% Evaluate average norm
avg_norm = mean(norm_values);
% Update bounds
if avg_norm > 10.^-8
max_u = mid_u;
else
min_u = mid_u;
end
% Check convergence
if abs(max_u - min_u) < tol
local_forcing_amp = mid_u;
iterations = iter;
break;
end
end
% If max iterations reached
if iter == maxIter
local_forcing_amp = (min_u + max_u) / 2;
iterations = maxIter;
end
% % Save mid_u values for analysis
% save('local_mid_u_bisection.mat', 'local_mid_u_bisection');
%
% % Separate data into two groups based on local_forcing_amp
above_threshold = local_mid_u_bisection > local_forcing_amp;
below_threshold = local_mid_u_bisection <= local_forcing_amp;
%
% save('local_u_bisection.mat',"local_u_bisection", "below_threshold", "above_threshold");
%
% Plot |u| as a function of time for both groups
colors = lines(16);
figure;
hold on;
for iter = 1:maxIter
for sim = 1:num_simulations
u = local_u_bisection{iter, sim};
if above_threshold(iter)
plot(tspan(late_time_indices), sqrt(sum(u(late_time_indices, :).^2, 2)), 'Color', colors(1, :)); % Above threshold in red
end
end
end
xlabel('Time');
ylabel('|u|');
title('|u| as a Function of Time');
legend('Above Threshold');
hold off;
figure;
hold on;
for iter = 1:maxIter
for sim = 1:num_simulations
u = local_u_bisection{iter, sim};
if below_threshold(iter)
plot(tspan(late_time_indices), sqrt(sum(u(late_time_indices, :).^2, 2)), 'Color', colors(1, :)); % Above threshold in red
end
end
end
xlabel('Time');
ylabel('|u|');
title('|u| as a Function of Time');
legend('Below Threshold');
hold off;
end
function [local_forcing_amp, iterations, local_u_bisection, local_mid_u_bisection] = bisectionMethodMulti2(min_u, max_u, tol, maxIter, A, B, tspan, u0, num_simulations, deltaf, delta_list, alpha,Beta,Gamma,KBG, KAG,KABG)
local_u_bisection = cell(length(maxIter), length(num_simulations)); % Preallocate cell array
local_mid_u_bisection = NaN * ones(length(maxIter), 1); % Preallocate array for mid_u values
% Modified Bisection Method with multiple simulations per step
for iter = 1:maxIter
mid_u = (min_u + max_u) / 2;
norm_values = zeros(num_simulations, 1);
% Run multiple simulations
for sim = 1:num_simulations
ini = randn(9, 1);
u0_sim = deltaf * ini / norm(ini);
u0_sim = 0*zeros(9,1);
[~, u, ~] = rk5_solver(@(t, u) ode_system_123(t, u, A, B, mid_u, alpha,Beta,Gamma,KBG, KAG,KABG), tspan, u0_sim);
norm_values(sim) = max(sqrt(sum(u.^2, 2))); % Max norm
% Store u for this simulation and iteration
local_u_bisection{iter, sim} = u;
end
% Store mid_u for this iteration
local_mid_u_bisection(iter) = mid_u;
% Evaluate average norm
avg_norm = mean(norm_values);
% Update bounds
if avg_norm > 10.^-8
max_u = mid_u;
else
min_u = mid_u;
end
% Check convergence
if abs(max_u - min_u) < tol
local_forcing_amp = mid_u;
iterations = iter;
return;
end
end
% If max iterations reached
local_forcing_amp = (min_u + max_u) / 2;
iterations = maxIter;
save('local_mid_u_bisection.mat', 'local_mid_u_bisection');
end