Lorenz 2 (Jordan NF) - VERICOMP

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Lorenz 2 (Jordan NF)

Equations:

$$\begin{align}\dot{x}_{0}&=11.8 \cdot x_{0} - x_{2} \cdot \left(0.29 \cdot x_{0} + 0.29 \cdot x_{1}\right)\\\dot{x}_{1}&=- 22.8 \cdot x_{1} + x_{2} \cdot \left(0.29 \cdot x_{0} + 0.29 \cdot x_{1}\right)\\\dot{x}_{2}&=- 2.67 \cdot x_{2} + \left(x_{0} + x_{1}\right) \cdot \left(2.2 \cdot x_{0} - 1.3 \cdot x_{1}\right)\end{align}$$

Description:

See Lorenz 1 (Classic Form); here the Jordan normal form exhibits more chaotic behavior for smaller values of the first initial condition.

Keywords:

non-periodic; chaotic; min test set

Plots:

Initial value problems:

98 (PINLBNU)
99 (PINLBNU)
100 (PINLBNU)
101 (PINLBU)
102 (PINLBU)
103 (PINLBU)

IVP 98

Class: PINLBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.92\\x_{1}(0)&=2.21\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=21.83830
c5: t_us=21.83660 , e_us=2.158241592553622e-06
c6: t_bd=3.942377 , e_bd=2.732430000000001e+02

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=74.46920
c5: t_us=74.46360 , e_us=2.128590004701891e-06
c6: t_bd=3.527398 , e_bd=2.698300000000000e+02

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=94.47980
c5: t_us=94.47310 , e_us=6.463660909616920e-06
c6: t_bd=3.467611 , e_bd=1.685071000000000e+02

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=144.21840
c5: t_us=144.20930 , e_us=5.157196845573253e-06
c6: t_bd=3.429101 , e_bd=1.137226000000000e+02

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04530
c5: t_us=0.04370 , e_us=1.069864197233983e-10
c6: t_bd=10.000000 , e_bd=3.990208661264205e-07

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07220
c5: t_us=0.06780 , e_us=4.624034488642792e-11
c6: t_bd=10.000000 , e_bd=1.709834585028602e-07

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10050
c5: t_us=0.09510 , e_us=3.229061462661775e-11
c6: t_bd=10.000000 , e_bd=1.200102648368784e-07

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.90840
c5: t_us=0.90180 , e_us=5.707606831606427e-09
c6: t_bd=10.000000 , e_bd=1.673225057530203e-05

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 98

IVP 99

Class: PINLBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.88\\x_{1}(0)&=2.21\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=21.72910
c5: t_us=21.72610 , e_us=2.778610088682854e-06
c6: t_bd=3.823955 , e_bd=2.728740000000000e+02

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=73.54210
c5: t_us=73.53760 , e_us=2.738893812193055e-06
c6: t_bd=3.916963 , e_bd=2.722830000000000e+02

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=93.53430
c5: t_us=93.52760 , e_us=8.541003829520832e-06
c6: t_bd=3.578477 , e_bd=1.626220000000000e+02

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=143.11870
c5: t_us=143.11010 , e_us=6.707411287720788e-06
c6: t_bd=3.502474 , e_bd=1.081718000000000e+02

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04520
c5: t_us=0.04400 , e_us=1.065778576503362e-10
c6: t_bd=10.000000 , e_bd=2.811467965102565e-07

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07380
c5: t_us=0.07040 , e_us=4.602540570886049e-11
c6: t_bd=10.000000 , e_bd=1.215462077652774e-07

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10250
c5: t_us=0.09900 , e_us=3.128164394183841e-11
c6: t_bd=10.000000 , e_bd=8.433948295305527e-08

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.91360
c5: t_us=0.90940 , e_us=5.617335929741785e-09
c6: t_bd=10.000000 , e_bd=1.173290695533069e-05

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 99

IVP 100

Class: PINLBNU

Initial conditions:

$$\begin{align}x_{0}(0)&=0.84\\x_{1}(0)&=2.21\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=21.40940
c5: t_us=21.40810 , e_us=5.608493154340977e-06
c6: t_bd=4.029383 , e_bd=4.956820000000000e+06

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=69.67020
c5: t_us=69.66440 , e_us=5.523746480662339e-06
c6: t_bd=3.857058 , e_bd=2.723720000000000e+02

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=88.86210
c5: t_us=88.85540 , e_us=1.732064328052729e-05
c6: t_bd=3.490272 , e_bd=1.630419000000000e+02

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=135.59250
c5: t_us=135.58420 , e_us=1.372035792712722e-05
c6: t_bd=3.455588 , e_bd=1.104661000000000e+02

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04610
c5: t_us=0.04430 , e_us=1.074269562195695e-10
c6: t_bd=10.000000 , e_bd=4.872699754798759e-07

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07460
c5: t_us=0.07150 , e_us=4.659561625430797e-11
c6: t_bd=10.000000 , e_bd=2.162089245416610e-07

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10210
c5: t_us=0.09820 , e_us=3.201172660283191e-11
c6: t_bd=10.000000 , e_bd=1.491558627719769e-07

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.90930
c5: t_us=0.90370 , e_us=5.540032432804765e-09
c6: t_bd=10.000000 , e_bd=2.038356115008355e-05

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 100

IVP 101

Class: PINLBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.915, 0.925\right]\\x_{1}(0)&\subset~\left[ 2.205, 2.215\right]\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.980374 , e_bd=6.420340000000000e+10

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=98.37000
c5: t_us=98.36440 , e_us=2.412551637614892e+02
c6: t_bd=1.001588 , e_bd=2.756170000000000e+02

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.904343 , e_bd=2.265700000000000e+09

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.882831 , e_bd=1.425938000000000e+20

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04750
c5: t_us=0.04480 , e_us=8.759972121643322e-02
c6: t_bd=2.118745 , e_bd=6.297061685185399e+03

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07650
c5: t_us=0.07310 , e_us=9.340663686844941e-02
c6: t_bd=2.104718 , e_bd=2.301463697712120e+04

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10620
c5: t_us=0.10180 , e_us=1.046812212522834e-01
c6: t_bd=2.055861 , e_bd=4.377582990602058e+04

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.92840
c5: t_us=0.92290 , e_us=8.226193734507348e-02
c6: t_bd=2.106333 , e_bd=4.360892256986240e+01

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 101

IVP 102

Class: PINLBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.875, 0.885\right]\\x_{1}(0)&\subset~\left[ 2.205, 2.215\right]\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.917816 , e_bd=2.941610000000000e+05

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.916984 , e_bd=6.230670000000000e+07

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.872376 , e_bd=1.019267000000000e+07

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.860835 , e_bd=4.638820000000000e+15

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04630
c5: t_us=0.04520 , e_us=8.806941132766077e-02
c6: t_bd=2.107223 , e_bd=6.250652240994053e+03

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07610
c5: t_us=0.07460 , e_us=9.386288811487198e-02
c6: t_bd=2.094041 , e_bd=2.250892123894743e+04

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10500
c5: t_us=0.10180 , e_us=1.055596827117071e-01
c6: t_bd=2.055636 , e_bd=4.063020414350792e+04

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.92890
c5: t_us=0.92520 , e_us=8.257766714405257e-02
c6: t_bd=2.098981 , e_bd=5.142295152365635e+01

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 102

IVP 103

Class: PINLBU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ 0.835, 0.845\right]\\x_{1}(0)&\subset~\left[ 2.205, 2.215\right]\\x_{2}(0)&=27.0\end{align}$$

Exact Solution^*:

No exact solution provided.

View test data:

Solver
Settings
Results

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.897170 , e_bd=3.211800000000000e+10

arithmetic: Interval
method: 7
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 1

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.896633 , e_bd=2.966330000000000e+05

arithmetic: Interval
method: 9
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.874247 , e_bd=1.220862000000000e+06

arithmetic: Interval
method: 11
abs. tolerance: 1e-12
initial stepsize: 1e-3
min. stepsize: 1e-5
monotonicity: 0

c4: t_c=999
c5: t_us=999 , e_us=999
c6: t_bd=0.871025 , e_bd=3.128660000000000e+28

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.04600
c5: t_us=0.04380 , e_us=8.847267888056898e-02
c6: t_bd=2.095618 , e_bd=6.311929177606109e+03

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.07590
c5: t_us=0.07320 , e_us=9.428937660949188e-02
c6: t_bd=2.087746 , e_bd=2.342507719576991e+04

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.10440
c5: t_us=0.10100 , e_us=1.063986980555498e-01
c6: t_bd=2.058427 , e_bd=4.201256966406120e+04

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.92580
c5: t_us=0.92090 , e_us=8.287296639172581e-02
c6: t_bd=2.082915 , e_bd=5.598629601822099e+01

Generate graphics:

DynIbex VNODE-LP

WPD for Problem 103

^* rounded (non-rigorously) to 5 decimal places to preserve screen real estate.