Flows

Flows shared by the mathematics packages: the Lorenz and Rössler flows.

LorenzElement(label, visual=None, *, x=1.0, y=1.0, z=1.0, sigma=10.0, rho=28.0, beta=8 / 3)

Bases: Element

Lorenz flow x' = sigma (y - x), y' = x (rho - z) - y, z' = x y - beta z.

A subclass that declares the states with a fractional StateSpec(order=q) gives the fractional Lorenz flow with Caputo derivatives of order q.

Parameters:

Name	Type	Description	Default
label	str	The label of the element.	required
visual	VisualSpec | None	How the element is drawn, or None for default_visual_.	None
x	Annotated[float, StateSpec()]	The initial convection intensity.	1.0
y	Annotated[float, StateSpec()]	The initial temperature difference between rising and falling currents.	1.0
z	Annotated[float, StateSpec()]	The initial deviation of the vertical temperature profile.	1.0
sigma	Annotated[float, ParameterSpec()]	The Prandtl number.	10.0
rho	Annotated[float, ParameterSpec()]	The Rayleigh number relative to its critical value.	28.0
beta	Annotated[float, ParameterSpec()]	The geometric factor.	8 / 3

Examples:

>>> from skcomplex.core.components.system import System
>>> from skcomplex.core.engine.runtime import Simulation
>>> from skcomplex.systems.mathematics.common.flows import LorenzElement
>>> system = System(elements=[LorenzElement('lorenz', x=0.0, y=0.0, z=2.0, beta=0.5)])
>>> trajectory = Simulation(system, dt=0.5).prepare().run(until=1.0).trajectory_
>>> [round(float(value), 4) for value in trajectory.values('lorenz.z')]
[2.0, 1.5576, 1.2131]

Source code in src/skcomplex/systems/mathematics/common/flows/_lorenz.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    x: Annotated[float, StateSpec()] = 1.0,
    y: Annotated[float, StateSpec()] = 1.0,
    z: Annotated[float, StateSpec()] = 1.0,
    sigma: Annotated[float, ParameterSpec()] = 10.0,
    rho: Annotated[float, ParameterSpec()] = 28.0,
    beta: Annotated[float, ParameterSpec()] = 8 / 3,
) -> None:
    super().__init__(label, visual=visual)
    self.x = x
    self.y = y
    self.z = z
    self.sigma = sigma
    self.rho = rho
    self.beta = beta

RosslerElement(label, visual=None, *, x=1.0, y=1.0, z=1.0, a=0.2, b=0.2, c=5.7)

Bases: Element

Rössler flow x' = -y - z, y' = x + a y, z' = b + z (x - c).

Parameters:

Name	Type	Description	Default
label	str	The label of the element.	required
visual	VisualSpec | None	How the element is drawn, or None for default_visual_.	None
x	Annotated[object, StateSpec()]	The initial first coordinate.	1.0
y	Annotated[object, StateSpec()]	The initial second coordinate.	1.0
z	Annotated[object, StateSpec()]	The initial third coordinate.	1.0
a	Annotated[object, ParameterSpec()]	The feedback of the second coordinate on itself.	0.2
b	Annotated[object, ParameterSpec()]	The constant drive of the third coordinate.	0.2
c	Annotated[object, ParameterSpec()]	The threshold of the third coordinate's growth.	5.7

Examples:

>>> from skcomplex.systems.mathematics.dynamical_systems import RosslerElement
>>> RosslerElement('rossler', c=4.0).c
4.0

Source code in src/skcomplex/systems/mathematics/common/flows/_rossler.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    x: Annotated[object, StateSpec()] = 1.0,
    y: Annotated[object, StateSpec()] = 1.0,
    z: Annotated[object, StateSpec()] = 1.0,
    a: Annotated[object, ParameterSpec()] = 0.2,
    b: Annotated[object, ParameterSpec()] = 0.2,
    c: Annotated[object, ParameterSpec()] = 5.7,
) -> None:
    super().__init__(label, visual=visual)
    self.x = x
    self.y = y
    self.z = z
    self.a = a
    self.b = b
    self.c = c