<p> rk4_2p(init,t0,t1,N) </p>

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The rkf4 2 parameters algorithm

Inside the rkf4  2 parammeters ... collapse in utilities init t0 t1 N rk4_2p U 1 el init : t 1 el t0 : h t1 t0 - N / : 13 mat j 1 N range n j : k1 h t n el U n el ω η D * eval : k2 h t n el h 2 / + U n el 12 / k1 * + ω η D * eval : k3 h t n el h 2 / + U n el 12 / k2 * + ω η D * eval : k4 h t n el h + U n el k3 + ω η D * eval : t n 1 + el t n el h + : U n 1 + el U n el 16 / k1 2 k2 * + 2 k3 * + k4 + (* + : 71 line for v U 1 el transpose : j 2 U rows range v v U j el transpose stack : for t v augment 51 line :

λ Y ω η D Y 2 el 1100 / 50 λ 4 / cos * ω Y 2 el * - η Y 1 el * - (* 21 mat : init 0121 mat : t0 0 : t1 80 : N 200 : ω 20 : η 100 : 16 mat 21 sys

rkf4ωη init t0 t1 N rk4_2p : Primitive rkf4ωη 1 col rkf4ωη 2 col augment : Derivative rkf4ωη 1 col rkf4ωη 3 col augment : 31 sys

Primitive Derivative 2 1 sys

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The source DE [NON homogeneous under forcing] t y t 2 diff 3 t * t y t diff * + 7 t y * - 4 t * ≡

Coefficients of the solved  NON-homogeneous DE A 13 tt * 7 - 4 tt * - 14 mat :

n expected number of solutions [Equation Power] n A cols 2 - :

i 2 : i n ≤ i i 1 + : f i 1 - el Ω i el : B i el Ω n i - 2 + el : 21 line for

n 2

fails
succeeds 1210 t sin - 14 mat 13 tt * 7 - 4 tt * - 14 mat 21 sys

B 0 Ω 2 el 21 mat

Construct the D(t,x) 
RK solving argument. B n 1 + el Ω 1 el : B n 2 + el 1 : f n el A B A 11 el / * (11 el - : 31 line

The 'f' RK  D(t,x)solving argument. f Ω 2 el 3 tt * Ω 2 el * - 7 Ω 1 el * + 4 tt * + 21 mat

<= conventionalise the RK solving argument tt Ω D f :

A Cauchy problem xx 1021 mat :

tt 1 el 0 : (start point ≡ T π : (segment length ≡ ε 104 -^: (accuracy ≡ h 0.25 : (step size ≡ 41 line

Метод Рунге-Кутта пятого порядка с автоматическим выбором шага

Metode Runge-Kutta orde lima dengan langkah adaptif

Fifth-order Runge–Kutta method with adaptive step accuracy ... adapted from Smath repository

Adaptive j tt 1 el h + : j T tt 1 el + h + ≤ j j h + : tx xx xx cols col : k1 h 3 / j tx D eval * : k2 h 3 / j h 3 / + tx k1 + D eval * : k3 h 3 / j h 3 / + tx k1 k2 + 2 / + D eval * : k4 h 3 / j h 2 / + tx 3 k1 * 9 k3 * + 8 / + D eval * : k5 h 3 / j h + tx 3 k1 * 9 k3 * - 2 / + 6 k4 * + D eval * : δ k1 92 / k3 * - 4 k4 * + 12 / k5 * - : δ min abs 5 ε * ≤ (δ max abs 5 ε * ≤ (& xx xx tx 12 / k1 4 k4 * + k5 + (* + augment : steps 1 xx cols el j : h h 2 * : 31 line j j h - : h h 2 / : 21 line if 81 line for steps transpose xx transpose augment 21 line :

<= ε & h are mutually compatible. Observe the number of values from tabulating "Adaptive".

assign to speed up Adaptive Adaptive :

Adaptive 1 col rows 43

idx Adaptive 1 col rows : 43

sub Adaptive 1 idx 13 submatrix : rk1 sub 1 col sub 2 col augment : rk2 sub 1 col sub 3 col augment : 31 sys

rk1 rk2 2 1 sys

t Ω D Ω 2 el 25 Ω 1 el * Ω 2 el + (* t sin - (- 21 mat : sol 0021 mat 03 π * 100 t Ω D Rkadapt : sol_1 sol 1 col sol 2 col augment : sol_2 sol 1 col sol 3 col augment : 41 sys

sol_1 sol_2 2 1 sys

A 2.5 - 101 - 1.25 22 mat :

A dn_LinAlgEigenvalues 7.014 8.264 - 21 mat

Unstable 2 - 11 - 122 mat

Decay 2 - 21 - 122 mat

Damped 2 - 31 - 122 mat 2 - 51 - 122 mat # 13 mat

Model the spiral in 'A' 2 - spiral 1 - zoom 22 mat orient A 11 el A 12 el / abs ≡ 21 line

t Y D A Y 1 el Y 2 el 21 mat * :

W 1 - 121 mat :

W 1021 mat 1121 mat 0121 mat # # 15 mat ≡

N 200 :

t Y D 54 Y 2 el * Y 1 el - (* 2 / 4 Y 1 el * - 5 Y 2 el * + 4 / 21 mat

U W 08 N t Y D rkfixed :

orient A 11 el A 12 el / abs : 0.25

Spiral U 2 col U 3 col augment :

Model the spiral space

Model the spiral: 1. increase length => 'spiral'. 2. expand in orinted xy => 'zoom'. 3. plot 'orient' support axis.

Spiral orient x * 2 1 sys

System components

U 1 col U 3 col augment U 1 col U 2 col augment 2 1 sys

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The 'μ' Logarithmic spiral

t X D μ X 1 el * X 2 el - X 1 el (2^X 2 el (2^+ (X 1 el * - μ X 2 el * X 1 el + X 1 el (2^X 2 el (2^+ (X 2 el * - 21 mat :

μ 0.2 - : t0 0 : t1 20 : N 1000 : 41 line

IC 0 0.5 21 mat :

Observe: D(t,X) is "algo" style & widow of 
the variable of integration 't'. Sol IC t0 t1 N var Deriv D rkfixed :

x Sol 1 col : X Sol 2 col : Y Sol 3 col : 31 line

U x X augment : V x Y augment : Lspiral Y X augment : 31 line

Best plot on square plot frame

Lspiral IC 2 el IC 1 el + 14 red 1 5 mat 2 1 sys

Phase components

U V 2 1 sys

A 2 - 81 - 122 mat : t S D A S 1 el S 2 el 21 mat * : 12 mat

N 200 :

Original:very slow 1 - 1 0.99 - range

A dn_LinAlgEigenvalues 6.694 7.694 - 21 mat

focus U 0 : U0 00013 mat : p 01 0.25 range W 1121 mat : U W 02 π * N t S D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 41 line :

W p 121 mat ≡

Focus focus 2 N 1 + (12 submatrix :

x y XY x y / - :

Focus Focus XY 2 1 sys

rkfixed 2 parameters . . .

... select a portion of either col(Sp,2) or col(Sp,3)

exercise the Genfit linfitCheby

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λ G ω η D G 2 el 1100 / 50 λ 4 / cos * ω G 2 el * - η G 1 el * - (* 21 mat : Sp 0121 mat 01001000 λ G 10100 D rkfixed : Sp1 Sp 1 col Sp 2 col augment : Sp2 Sp 1 col Sp 3 col augment : 41 sys

Sp1 Sp2 2 1 sys

collect multiple solutions wrt either parameter

mrkf=> multisolutions. This program collects the 's' solutions over 'j' counter.
Collector is for fixed 'η'. It can be reversed collect 'η' for fixed 'ω' ...
The counter 'j' collects 'ω' from 'list', that can be expanded/reduced, each
'ω' can be modified at will ... a very impressive Smath functionality. 
Program is auto-coded for downstream user, i.e: can be collapsed in utilities. V list c mrkf j 1 : init V 1 el : t0 V 2 el : t1 V 3 el : N V 4 el : ω V 5 el : η V 4 el : 16 mat X is the discretised independent variate domain X t0 t1 t0 t1 t0 - (N / + range : solve and collect 'ω' solutions in horizontal nest ω list sol 1 j el init t0 t1 N λ Φ ω η D rkfixed c col : j j 1 + : 21 line for unest the solutions in contiguous data table v sol 1 el : j 2 sol cols range v v sol j el augment : for v 31 line X v augment 91 line :

<= Φ

The project is all specified/contained in there.
mrkf(V,list,c) takes your demand. Avoid shadow
in context wrt variable of integration => 'λ' λ Φ ω η D Φ 2 el 1100 / 50 λ 4 / cos * ω Φ 2 el * - η Φ 1 el * - (* 21 mat : specify range integers V 0121 mat 040100 ω 10016 mat : list 5206010014 mat : 41 sys

Export 'list' => c[2 solution] ... c[3 derivative] V list 3 mrkf 01111 0.4 1.1 1.04 0.89 0.77 0.8 1.02 0.91 0.68 0.51 1.2 0.78 0.65 0.42 0.27 1.6 0.42 0.31 0.15 0.07 2 0.01 - 0.06 - 0.09 - 0.08 - 2.4 0.43 - 0.39 - 0.28 - 0.17 - 2.8 0.79 - 0.66 - 0.39 - 0.22 - 3.2 1.03 - 0.81 - 0.44 - 0.24 - 95 mat

Primitive
Derivative Pr V list 2 mrkf : Dr V list 3 mrkf : 21 sys

select 'ω' plot from list. 'ω' starts @ 2

plot Pr 1 col Pr ω col augment Dr 1 col Dr ω col augment 21 sys :

plot

A 3 - 113 - 22 mat : t K D A K 1 el K 2 el 21 mat * : 12 mat

N 100 : span 1 : orient 1 : 31 line

A dn_LinAlgEigenvalues 2 - 4 - 21 mat

knot_1 U0 01113 mat : p 1 - 1 0.75 - range s p span 21 mat : kn s 02 N t K D rkfixed : U0 U0 kn stack : 31 line for U0 3 col U0 2 col augment 31 line :

knot_2 U0 01113 mat : p 1 - 1 0.75 - range s p span - 21 mat : kn s 02 N t K D rkfixed : U0 U0 kn stack : 31 line for U0 3 col U0 2 col augment 31 line :

Stable node

knot_1 knot_2 orient x * 3 1 sys

start 406 :

K1 knot_1 start start 100 + (12 submatrix : K2 knot_2 start start 100 + (12 submatrix : 21 line

Easy to collect more pairs to immitate the "stable node" continuous trace from book.

K1 K2 2 1 sys

Pμ

The parametric Rkadapt

t Pα α D α 1 el Pα 2 el Pα 1 el - (* Pα 1 el α 3 el Pα 3 el - (* Pα 1 el - Pα 1 el Pα 2 el * α 2 el Pα 3 el * - 31 mat :

init 0.1 0.1 0.1 31 mat :

t.0 0 :

t.max 50 :

n 2000 :

res init t.0 t.max n t Pα 2 1.5 2031 mat D Rkadapt :

X res 1 col :

Y res 2 col :

Z res 3 col :

Y Z augment

res init t.0 t.max n t Pα 21531 mat D Rkadapt :

X res 1 col :

Y res 2 col :

Z res 3 col :

Y Z augment

res 1012 -^1012 -^1012 -^31 mat t.0 t.max n t Pα 21531 mat D Rkadapt :

x y axes x y / - :

X res 1 col :

Y res 2 col :

Z res 3 col :

Y Z augment axes 2 1 sys