Flows with Strange Attractors

dx/dt = σ(y - x)

dy/dt = x(ρ - z) - y

dz/dt = xy - βz

σ = 10

ρ = 28

β = 8/3

dy/dt = x(ρ - z) - y

dz/dt = xy - βz

σ = 10

ρ = 28

β = 8/3

dx/dt = - y - z

dy/dt = x + ay

dz/dt = b + z(x - c)

a = 0.1

b = 0.1

c = 14

dy/dt = x + ay

dz/dt = b + z(x - c)

a = 0.1

b = 0.1

c = 14

x_{n+1} = y_{n} + 1 - ax_{n}^{2}

y_{n+1} = bx_{n}

(This is a 2-D attractor, so there is no z coordinate, technically)

a = 1.4

b = 0.3

y

(This is a 2-D attractor, so there is no z coordinate, technically)

a = 1.4

b = 0.3

dx/dt = α(y - x - f(x))

dy/dt = x - y + z

dz/dt = - βy

f(x) = m_{1}x + (m_{0}-m_{1})/2 * |x+1| - |x-1|)

The equation for f(x) was obtained from matlab code provided here

α = 15.6

β = 28

m_{0} = -1.143

m_{1} = -0.714

dy/dt = x - y + z

dz/dt = - βy

f(x) = m

The equation for f(x) was obtained from matlab code provided here

α = 15.6

β = 28

m

m

dx/dt = y

dy/dt = μ(1 - x^{2})y - x

This is a 2-D oscillator. I showed flow for various random values of μ. It also looks nice if you set z = μ

dy/dt = μ(1 - x

This is a 2-D oscillator. I showed flow for various random values of μ. It also looks nice if you set z = μ

dx/dt = y(z - 1 + x^{2}) + γx

dy/dt = x(3z + 1 - x^{2}) + γy

dz/dt = - 2z(α + xy)

α = 0.98

γ = 0.1

dy/dt = x(3z + 1 - x

dz/dt = - 2z(α + xy)

α = 0.98

γ = 0.1

x_{n+1} = x_{n}^{2} - y_{n}^{2} + ax_{n} + by_{n}

y_{n+1} = 2x_{n}y_{n} + cx_{n} + dy_{n}

(This is a 2-D dynamical system, so there is no z coordinate, technically)

a = 0.9

b = -0.6013

c = 2

d = 0.5

y

(This is a 2-D dynamical system, so there is no z coordinate, technically)

a = 0.9

b = -0.6013

c = 2

d = 0.5

x_{n+1} = 1 + u(x_{n}cos(t_{n}) - y_{n}sin(t_{n}))

y_{n+1} = u(x_{n}sin(t_{n}) + y_{n}cos(t_{n}))

t_{n} = 0.4 - 6/(1 + x_{n}^{2} + y_{n}^{2})

(This is a 2-D dynamical system, so there is no z coordinate, technically)

u = 0.85

y

t

(This is a 2-D dynamical system, so there is no z coordinate, technically)

u = 0.85

dx/dt = yz

dy/dt = xz

dz/dt = xy

Chris Rainey is a former REU/illiMATH student who made up his own set of ODEs for a dynamical system

dy/dt = xz

dz/dt = xy

Chris Rainey is a former REU/illiMATH student who made up his own set of ODEs for a dynamical system

dx/dt = α(y + x)/z

dy/dt = x(β - x - y - z)

dz/dt = xyz/γ

α = 18

β = 42

γ = 15

My own creation (a reward for hard work)

dy/dt = x(β - x - y - z)

dz/dt = xyz/γ

α = 18

β = 42

γ = 15

My own creation (a reward for hard work)