Examples

Basic examples for each kind of system, ready to try. “Try” opens them with the input filled in.

Questions ready to try

System	Field	s	t
Lyness map (a = 3)	𝔽_1000003	(2,5)	(7,11)	Try
Euler top (Kahan's method) (a1 = 1, a2 = 2, a3 = -3, h = 1/2)	𝔽_2147483647	(1,2,3)	(4,5,6)	Try
cat map	𝔽_1009	(1,0)	(2,1)	Try
Euler top	ℚ, B=10	(1/3,1/5,1/7)	(1/3,1/5,1/7)	Try
Henon (a = 5)	𝔽_1009	(1,2)	(3,4)	Try
Lyness type (b = 3)	𝔽_1009	(2,5)	(7,11)	Try

Example system files

affine cat map

Try

name: affine cat map
vars: x, y
map:
  x' = 2*x + y + 1
  y' = x + y

anharmonic oscillator in the plane

Try

name: anharmonic oscillator in the plane
vars: x, y
params: k = 1/10
lagrangian: L = (x' - x)^2/2 + (y' - y)^2/2 - k*(((x + x')/2)^2 + ((y + y')/2)^2)^2
symmetry:
  so: x^2 + y^2
observables:
  J = x*y' - y*x'

cat map

Try

name: cat map
vars: x, y
map:
  x' = 2*x + y
  y' = x + y

E2 step, written out

Try

name: E2 step, written out
vars: p1, p2, x1, x2
map:
  p1' = x1
  p2' = x2
  x1' = (2*(x1*p1 - 2*x2*p2)*x1 - (x1^2 - 2*x2^2)*p1)/(p1^2 - 2*p2^2)
  x2' = (2*(x1*p1 - 2*x2*p2)*x2 - (x1^2 - 2*x2^2)*p2)/(p1^2 - 2*p2^2)

E2 geodesic

Try

name: E2 geodesic
vars: x, y, z
params: a = 0
potential: -log(x^2 - y^2 - z^2)
alpha: a
symmetry:
  so: x^2 - y^2 - z^2
observables:
  Q = x'^2 - y'^2 - z'^2

Euler top

Try

name: Euler top
vars: x1, x2, x3
params: a1 = 1, a2 = 2, a3 = -3
field:
  dx1/dt = a1*x2*x3
  dx2/dt = a2*x3*x1
  dx3/dt = a3*x1*x2
kahan: h = 1/2

Euler top, expanded

Try

name: Euler top, expanded
vars: x, y, z
params: a = 1, b = 2, c = -3, h = 1/2
map:
  x' = ((1 - b*c*h^2*x^2)*x + a*h*(c*h*x*y + z)*y + a*h*(b*h*x*z + y)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)
  y' = (b*h*(c*h*x*y + z)*x + (1 - a*c*h^2*y^2)*y + b*h*(a*h*y*z + x)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)
  z' = (c*h*(b*h*x*z + y)*x + c*h*(a*h*y*z + x)*y + (1 - a*b*h^2*z^2)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)

Euler top, other coordinates

Try

name: Euler top, other coordinates
vars: u, v, w
field:
  du/dt = 4*u**2 - 15*u*v - 26*u*w + 19*u + 14*v**2 + 49*v*w - 35*v + 42*w**2 - 63*w + 21
  dv/dt = 2*u**2 - 4*u*v - 6*u*w + 6*u - 2*v*w - 2*v - 4*w**2 - 2*w + 2
  dw/dt = -2*u*v - 4*u*w + 2*u + 4*v**2 + 15*v*w - 9*v + 14*w**2 - 17*w + 5
kahan: h = 1/3

Henon

Try

name: Henon
vars: x, y
params: a = 2, b = 3
map:
  x' = y
  y' = a - b*x - y^2
inverse:
  x' = (a - x^2 - y)/b
  y' = x

Lotka-Volterra (Kahan)

Try

name: Lotka-Volterra (Kahan)
vars: x, y
field:
  dx/dt = x - x*y
  dy/dt = x*y - y
kahan: h = 1/5

Lyness, other coordinates

Try

name: Lyness, other coordinates
vars: x, y
map:
  x' = (-x - y + (x + y - 5)*(2*x + y + 1) - 1)/(2*x + y + 1)
  y' = (2*x + 2*y + (-x - y + 7)*(2*x + y + 1) + 2)/(2*x + y + 1)

Lyness type

Try

name: Lyness type
vars: x, y
params: a = 3, b = 2
map:
  x' = y
  y' = (a + b*y)/x

discrete oscillator

Try

name: discrete oscillator
vars: x
params: h = 1/3
lagrangian: L = (x' - x)^2/2 - h^2*(x^2 + x'^2)/4

parabolic shear

Try

name: parabolic shear
vars: x, y
map:
  x' = 2*x - y + 3
  y' = x + 1

quartic midpoint

Try

name: quartic midpoint
vars: x
params: h = 1/2
lagrangian: L = (x' - x)^2/2 - h^2*((x + x')/2)^4/4

two generators

Try

name: two generators
vars: x, y
generator r:
  x' = -y
  y' = x
generator s:
  x' = x + y^2
  y' = y

Loading…

Settings that existing methods cannot answer, or answer only with great difficulty: fields so large that following the orbit would take years, questions over ℚ that following the orbit can never settle, and known systems hidden in other coordinates. Every answer here was computed with Structure Computer and confirmed by the independent checker; the times are for Id(s, t) alone, measured on our own PC. “Try” opens an example with its input filled in; some of the other questions computed there take several minutes.

1. Lyness map · 𝔽_q, q = 2⁶¹ − 1
A state 7.8 quadrillion steps away
Try
System
Lyness map (a = 3)
s
(2, 5)
t
(813066766938016773, 992646061924517494)
Following the orbit step by step
Following the orbit one step at a time takes 7,863,588,611,074,867 steps: about three months even at a billion steps a second, and hundreds of years in a general-purpose computer algebra system. To add “and no other step count reaches t”, the whole period (576,460,752,629,115,698 steps, about 70 times as many) would have to be followed too.
Structure Computer
Without following the orbit, it answers from theorems proved in Lean: every step count that reaches t, and that no other count does, with a certificate.
t = σⁿ s ⇔ n ≡ 7,863,588,611,074,867 (mod 576,460,752,629,115,698)
Certain, checked · 0.2 s
2. Lyness map · 𝔽_q, q = 2⁶¹ − 1
400 quadrillion steps along an orbit that breaks off
Try
System
Lyness map (a = 3)
s
(652450593252319644, 397870522539971473)
t
(557047722706213665, 1530465154515367894)
Following the orbit step by step
Going on from s, the orbit somewhere reaches a state where the formula divides by zero and breaks off (s is not regular). Finding where, or whether t comes first, takes more than 10¹⁷ steps if the orbit is followed: over ten years at a billion steps a second.
Structure Computer
It proves that s is not regular without following the orbit, and answers that before the orbit breaks off t is reached in exactly 403,567,694,157,578,575 steps, and in no other number of steps.
s is not regular (t is reached in 403,567,694,157,578,575 steps)
Certain, checked · 10 s
3. Lyness map · 𝔽_q, q = 2⁶¹ − 1
On the same curve, yet never reached
Try
System
Lyness map (a = 3)
s
(1473736884130421033, 1913329524460333331)
t
(1744307874685056163, 461811991117244972)
Following the orbit step by step
t has the same value of the conserved quantity as s (it lies on the same curve), so comparing conserved quantities cannot tell them apart. To confirm “never” by following the orbit, the whole orbit of s has to be gone round: 192,153,584,192,171,886 steps, about six years at a billion steps a second.
Structure Computer
It answers, with a certificate resting on theorems, that no number of steps reaches t.
no path
Certain, checked · 0.2 s
4. Lyness map · ℚ
Over ℚ, (3, 7) never reaches (7, 3)
Try
System
Lyness map (a = 3)
s
(3, 7)
t
(7, 3)
Following the orbit step by step
Both states are positive and have the same value of the conserved quantity. Over ℚ the orbit can come back only after a multiple of 3,240,671,700 steps (Structure Computer answers that too), so however far it is followed without meeting t, t may still come later: neither numerical computation nor computer algebra can conclude “never” by following it. The similar pair (2, 5) and (5, 2) is joined by going 3 steps back.
Structure Computer
It proves that no number of steps over ℚ reaches t, with a certificate resting on theorems proved in Lean.
no path
Certain, checked · 0.1 s
5. Euler top (Kahan's method) · 𝔽_q, q = 1,099,511,627,791 ≈ 2⁴⁰
Euler top: 137 billion steps away
Try
System
Euler top (Kahan's method) (a1 = 1, a2 = 2, a3 = -3, h = 1/2)
s
(3, 1, 4)
t
(3, -1, -4)
Following the orbit step by step
Following this map of three variables one step at a time takes 137,438,890,526 steps: more than a day in a general-purpose computer algebra system. To add “and no other step count”, the whole period (274,877,781,052 steps) would have to be followed too.
Structure Computer
Without following the orbit, it answers from theorems that t, the state with two signs flipped, appears exactly half a period after s.
t = σⁿ s ⇔ n ≡ 137,438,890,526 (mod 274,877,781,052)
Certain, checked · 0.2 s
6. Euler top (Kahan's method) · ℚ
Euler top over ℚ: the same pair is never joined
Try
System
Euler top (Kahan's method) (a1 = 1, a2 = 2, a3 = -3, h = 1/2)
s
(3, 1, 4)
t
(3, -1, -4)
Following the orbit step by step
The pair of the previous example, now over ℚ. The two conserved quantities have the same values at both states, and the orbit can come back only after a multiple of 575,827,964 steps, so following it never settles the question.
Structure Computer
It proves that over ℚ no number of steps reaches t, although the two states are joined over the finite field of the previous example.
no path
Certain, checked · 0.2 s
7. Euler top (Kahan's method) · 𝔽_q, q = 1,099,511,627,791 ≈ 2⁴⁰
Euler top: the same conserved quantities, yet never reached
Try
System
Euler top (Kahan's method) (a1 = 1, a2 = 2, a3 = -3, h = 1/2)
s
(3, 1, 4)
t
(-3, -1, 4)
Following the orbit step by step
(3, 1, 4) and (−3, −1, 4) have the same values of both conserved quantities, so comparing them cannot tell the states apart. To confirm “never” by following the orbit, the whole orbit of s has to be gone round: 274,877,781,052 steps, more than three days in a general-purpose computer algebra system.
Structure Computer
It answers, with a certificate resting on theorems, that no number of steps reaches t, whereas in the fifth example the state with two other signs flipped is reached at exactly half the period.
no path
Certain, checked · 0.2 s
8. Lyness map, disguised · 𝔽_q, q = 2⁶¹ − 1
Seeing through an unfamiliar rational map
Try
System
Show the system file
name: a rational map of the plane vars: x, y map: x' = -(3*x**2 - 5*x*y + x + 2*y**2 - 3*y - 2)/(3*x - 2*y + 5) y' = -(3*x**2 - 5*x*y - 19*x + 2*y**2 + 10*y - 33)/(3*x - 2*y + 5)
s
(21, 33)
t
(492515881573357819, 1485161943497875320)
Following the orbit step by step
Read as formulas, it is an ordinary-looking rational map: quadratics divided by a linear form. Unless one notices that it is a known system, the only way is to follow the orbit, 7,863,588,611,074,867 steps.
Structure Computer
It sees that this is the Lyness map in other coordinates, and answers from theorems as in the first example (the step counts are the same). On the Try page, the conserved quantity is also found as a formula in these coordinates (Q3, Q4).
t = σⁿ s ⇔ n ≡ 7,863,588,611,074,867 (mod 576,460,752,629,115,698)
Certain, checked · 0.2 s
9. Lyness map, disguised · ℚ
An unfamiliar rational map: “never” over ℚ
Try
System
Show the system file
name: a rational map of the plane vars: x, y map: x' = -(3*x**2 - 5*x*y + x + 2*y**2 - 3*y - 2)/(3*x - 2*y + 5) y' = -(3*x**2 - 5*x*y - 19*x + 2*y**2 + 10*y - 33)/(3*x - 2*y + 5)
s
(26, 40)
t
(22, 32)
Following the orbit step by step
The rational map of the previous example, asked over ℚ. Nothing in the formulas shows that a conserved quantity exists, and the orbit can come back only after a multiple of 3,240,671,700 steps, so following it, however far, can never conclude “never”.
Structure Computer
It sees that this is the Lyness map in other coordinates, and proves, with a certificate resting on theorems proved in Lean, that no number of steps over ℚ reaches t (the fourth example, asked in other coordinates).
no path
Certain, checked · 0.1 s
10. Euler top, disguised · 𝔽_q, q = 1,099,511,627,791 ≈ 2⁴⁰
The identity of a quadratic field full of coefficients
Try
System
Show the system file
name: a quadratic field in three variables (Kahan) vars: u, v, w field: du/dt = -5*u**2/9 + 7*u*v/9 + 4*u*w/9 + 5*u/3 - 2*v**2/9 - v*w/9 - 5*v/3 + w**2/9 - 5*w/3 dv/dt = -10*u**2/9 + 2*u*v/9 + 5*u*w/9 + 3*u + 8*v**2/9 - 14*v*w/9 + 6*v + 5*w**2/9 - 6*w + 9 dw/dt = -13*u**2/9 - 13*u*v/9 + 8*u*w/9 + 5*u/3 + 26*v**2/9 - 23*v*w/9 + 49*v/3 + 5*w**2/9 - 23*w/3 + 18 kahan: h = 1/2
s
(6, 4, 14)
t
(4, -6, -2)
Following the orbit step by step
Kahan's discretisation of a vector field in three variables whose every component is a full quadratic polynomial with fractional coefficients. Unless one notices that it is a known system, the only way is to follow the orbit, 137,438,890,526 steps.
Structure Computer
It sees that this is the Euler top in other coordinates, and answers from theorems with the same step counts as the Euler top example above. On the Try page, the two conserved quantities are also found as formulas in these coordinates (Q3, Q4).
t = σⁿ s ⇔ n ≡ 137,438,890,526 (mod 274,877,781,052)
Certain, checked · 0.2 s

Questions ready to try

System	Field	s	t
Lyness map (a = 3)	𝔽_1000003	(2,5)	(7,11)	Try
Euler top (Kahan's method) (a1 = 1, a2 = 2, a3 = -3, h = 1/2)	𝔽_2147483647	(1,2,3)	(4,5,6)	Try
cat map	𝔽_1009	(1,0)	(2,1)	Try
Euler top	ℚ, B=10	(1/3,1/5,1/7)	(1/3,1/5,1/7)	Try
Henon (a = 5)	𝔽_1009	(1,2)	(3,4)	Try
Lyness type (b = 3)	𝔽_1009	(2,5)	(7,11)	Try

Example system files

affine cat map

Try

name: affine cat map
vars: x, y
map:
  x' = 2*x + y + 1
  y' = x + y

anharmonic oscillator in the plane

Try

name: anharmonic oscillator in the plane
vars: x, y
params: k = 1/10
lagrangian: L = (x' - x)^2/2 + (y' - y)^2/2 - k*(((x + x')/2)^2 + ((y + y')/2)^2)^2
symmetry:
  so: x^2 + y^2
observables:
  J = x*y' - y*x'

cat map

Try

name: cat map
vars: x, y
map:
  x' = 2*x + y
  y' = x + y

E2 step, written out

Try

name: E2 step, written out
vars: p1, p2, x1, x2
map:
  p1' = x1
  p2' = x2
  x1' = (2*(x1*p1 - 2*x2*p2)*x1 - (x1^2 - 2*x2^2)*p1)/(p1^2 - 2*p2^2)
  x2' = (2*(x1*p1 - 2*x2*p2)*x2 - (x1^2 - 2*x2^2)*p2)/(p1^2 - 2*p2^2)

E2 geodesic

Try

name: E2 geodesic
vars: x, y, z
params: a = 0
potential: -log(x^2 - y^2 - z^2)
alpha: a
symmetry:
  so: x^2 - y^2 - z^2
observables:
  Q = x'^2 - y'^2 - z'^2

Euler top

Try

name: Euler top
vars: x1, x2, x3
params: a1 = 1, a2 = 2, a3 = -3
field:
  dx1/dt = a1*x2*x3
  dx2/dt = a2*x3*x1
  dx3/dt = a3*x1*x2
kahan: h = 1/2

Euler top, expanded

Try

name: Euler top, expanded
vars: x, y, z
params: a = 1, b = 2, c = -3, h = 1/2
map:
  x' = ((1 - b*c*h^2*x^2)*x + a*h*(c*h*x*y + z)*y + a*h*(b*h*x*z + y)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)
  y' = (b*h*(c*h*x*y + z)*x + (1 - a*c*h^2*y^2)*y + b*h*(a*h*y*z + x)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)
  z' = (c*h*(b*h*x*z + y)*x + c*h*(a*h*y*z + x)*y + (1 - a*b*h^2*z^2)*z) / (1 - h^2*(b*c*x^2 + c*a*y^2 + a*b*z^2) - 2*h^3*a*b*c*x*y*z)

Euler top, other coordinates

Try

name: Euler top, other coordinates
vars: u, v, w
field:
  du/dt = 4*u**2 - 15*u*v - 26*u*w + 19*u + 14*v**2 + 49*v*w - 35*v + 42*w**2 - 63*w + 21
  dv/dt = 2*u**2 - 4*u*v - 6*u*w + 6*u - 2*v*w - 2*v - 4*w**2 - 2*w + 2
  dw/dt = -2*u*v - 4*u*w + 2*u + 4*v**2 + 15*v*w - 9*v + 14*w**2 - 17*w + 5
kahan: h = 1/3

Henon

Try

name: Henon
vars: x, y
params: a = 2, b = 3
map:
  x' = y
  y' = a - b*x - y^2
inverse:
  x' = (a - x^2 - y)/b
  y' = x

Lotka-Volterra (Kahan)

Try

name: Lotka-Volterra (Kahan)
vars: x, y
field:
  dx/dt = x - x*y
  dy/dt = x*y - y
kahan: h = 1/5

Lyness, other coordinates

Try

name: Lyness, other coordinates
vars: x, y
map:
  x' = (-x - y + (x + y - 5)*(2*x + y + 1) - 1)/(2*x + y + 1)
  y' = (2*x + 2*y + (-x - y + 7)*(2*x + y + 1) + 2)/(2*x + y + 1)

Lyness type

Try

name: Lyness type
vars: x, y
params: a = 3, b = 2
map:
  x' = y
  y' = (a + b*y)/x

discrete oscillator

Try

name: discrete oscillator
vars: x
params: h = 1/3
lagrangian: L = (x' - x)^2/2 - h^2*(x^2 + x'^2)/4

parabolic shear

Try

name: parabolic shear
vars: x, y
map:
  x' = 2*x - y + 3
  y' = x + 1

quartic midpoint

Try

name: quartic midpoint
vars: x
params: h = 1/2
lagrangian: L = (x' - x)^2/2 - h^2*((x + x')/2)^4/4

two generators

Try

name: two generators
vars: x, y
generator r:
  x' = -y
  y' = x
generator s:
  x' = x + y^2
  y' = y

Examples

Questions ready to try

Example system files

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Following the orbit step by step

Structure Computer

Questions ready to try

Example system files