Rotary states

The minimalistic framework for finding rotational regimes and determining their stability

Consider a system of autonomous ODE that describes $N$ coupled phase oscillators:

$$\dot{X} = F(X), \ X \in R^n$$

where $X$ is state vector of coupled phase oscillators. Without losing generality, we can write $X$ as follows:

$$X = (\varphi_1, \dot{\varphi}_1, \varphi_2, \dot{\varphi}_2, ... , \varphi_N, \dot{\varphi}_N).$$

This framework allows you to find rotational regimes and determine their stability.

My master's thesis

This presentation is about my master's thesis based on this library

Rotational regimes finding

The rotation modes are described by the rotation period, the phase period and the initial conditions.

• rotation period $\in R$
• phase period $= 2\pi \cdot k, \ k \in Z$
• initial conditions $\in R^n$

If you want to find some rotational mode, you need to have a few things:

• Approximate rotation period: $T_0 \in R$
• Approximate initial conditions: $IC_0=(\varphi_{1_0}, \dot{\varphi}_{1_0}, \varphi_{2_0}, \dot{\varphi}_{2_0}, ... , \varphi_{N_0}, \dot{\varphi}_{N_0})$
• Phase period: $phase\_period = 2\pi \cdot k$

Without losing generality for approximate initial conditions, we can set the $\varphi_{1_0} = 0$ and this is required for $IC_0$

Code snippet for finding $4\pi$ rotary states in system $\dot{X} = F(X)$:

from rotary_states import limit_cycles
# define 'IC_0', 'T_0', 'F', 'args'
assert IC_0[0] == 0

# 'IC_0' - Approximate initial conditions
# 'T_0' - Approximate rotation period
# 'F' - Right side of system
# 'args' - A tuple of constants used in 'F'
# 'IC' - Initial conditions we found
# 'T' - Rotation period we found

T, IC = limit_cycles.find_limit_cycle(F, args, IC_0, T_0, phase_period=4*mt.pi)

For example, we will consider a chain of identic coupled oscillators with inertia $m$, friction $\lambda$ and constant rotational moment $\gamma$:

$$\ddot{\varphi}_i + \lambda \dot{\varphi}_i + \sin{\varphi}_i = \gamma + k \left[ \sin(\varphi_{i+1} - \varphi_i) + \sin(\varphi_{i-1} - \varphi_i) \right].$$

There are a lot of some rotational regimes you can find in that system. Also you can see the our previous publication about this study, but in this example you can see how we can find some $4\pi$-periodic rotational regime with specific parameters in that system.

Determination of the stability of the rotational mode

To determine the stability of the rotational mode we can use the Floquet theory.

First of all we have to linearize the system around rotational regime we interested in by replacement:

$$\varphi_i = \delta\varphi_i + \psi_i,$$

we get the following system:

$$\dot{\delta\varphi} = A(t)\delta\varphi,$$

where $A(t) = A(t + T)$ - periodic matrix.

That system describes the perturbations. Stability of zero solution $\delta \varphi_i$ determine stability of rotational regime $\psi_i$.

The eigenvalues of Monodromy matrix are determine zero solution stability of $\delta\varphi$.

If we have rotation period, phase period and initial conditions of the rotational regime we are interested in, we can determine their stability a that way:

from rotary_states import limit_cycles
from numpy.linalg import eig
# 'IC' - Initial conditions
# 'T' - Rotation period
# 'F' - Right side of system
# 'F_linear' - A right side of system linearized around the rotational regime
# 'args' - A tuple of constants used in 'F'
# 'args_linear' - A tuple of constants used in 'F_linear'
# 'args' - A tuple of constants used in 'F'
# 'M' - Monodromy matrix we found.

M = limit_cycles.get_monogrommy_matrix(F, F_linear, IC, T, args_linear, args)

# Eigenvalues
eigenvalues, _ = eig(M)

# Stability
if (np.absolute(eigenvalues) < 1).all():
    print("STABLE")
else:
    print("UNSTABLE")

See the example here

Setup environment

We use poetry to manage dependencies

Poetry install:

pip3 install poetry

Install dependencies

# in root of this dir
poetry install

Run tests

# in root of this dir
poetry run pytest