79211195-ac16-42e7-8206-6d5cf2a27755 12 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Option 2: the RK solving is given otherwise, generally from source or from user experiment.

Van der Pol t X D X 2 el 1 X 1 el (2^- (X 2 el * X 1 el - 21 mat :

Cauchy initial conditions IC 1021 mat :

tx IC IC cols col : 1021 mat

t 1 el 0 : (start point ≡ T 10 : (segment length ≡ ε 106 -^: (a5ccuracy ≡ h 0.1 : (step size ≡ 41 line

<= ε & h are mutually compatible. Observe the number of values.

j t 1 el h + : j T t 1 el + h + ≤ j j h + : tx IC IC 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 * ≤ (& IC IC tx 12 / k1 4 k4 * + k5 + (* + augment : steps 1 IC cols el j : h h 2 * : 31 line j j h - : h h 2 / : 21 line if 81 line for

after the for-loop has resumed tx IC IC cols col : 1.526 - 0.771 21 mat

result steps IC stack transpose :

Number of values result rows 223

Prepare/extract solutions X steps transpose : RK_1 X result 2 col augment : RK_2 X result 3 col augment : 31 line

Discrete export of solution(s).

RK_1 RK_2 2 1 sys

x sol_1 0 x ≤ (x T ≤ (& RK_1 1 col RK_1 2 col x cinterp out range if :

x sol_2 0 x ≤ (x T ≤ (& RK_2 1 col RK_2 2 col x cinterp out range if :

cycle result 2 col result 3 col augment :

Van der Pol fine meshing over the canvas plot.

x sol_1 x sol_2 2 1 sys

Van der Pol cycle

cycle IC 1 1 el IC 2 1 el . 14 red 1 5 mat 2 1 sys

Lorentz attractor Smath Rkadapt built-in

u Y D 8 Y 1 el * - 8 Y 2 el * + 30 Y 1 el * Y 1 el + Y 1 el Y 3 el * - Y 1 el Y 2 el * 83 / Y 3 el * - 31 mat :

t0 0 : t1 10 : N 999 : 31 line

init 1 - 0131 mat :

s init t0 t1 N u Y D Rkadapt :

s rows 1000

y0 s 1 col s 2 col augment : y1 s 1 col s 3 col augment : y2 s 1 col s 4 col augment : 31 line

Lorenz triplet solutions

y0 y1 y2 3 1 sys

Lorentz s 2 col s 3 col augment : θ rot θ cos θ sin - θ sin θ cos 22 mat : θ π - 4 / : Rot Lorentz θ rot * : 41 line

Lorentz 0 Rot 2 1 sys