phys_581.assignment_4 ===================== .. py:module:: phys_581.assignment_4 .. autoapi-nested-parse:: Assignment 4: Chaos Functions --------- .. autoapisummary:: phys_581.assignment_4.compute_lyapunov Module Contents --------------- .. py:function:: compute_lyapunov(compute_dy_dt, y0, dy0=None, t0=0, dt=None, min_norm=None, Nsamples=None, norm=np.linalg.norm, rng=_DEFAULT_RNG, debug=False, solve_ivp_args=None) Return a list of uncorrelated values `lams` estimating the maximal Lyapunov exponent for the ODE. The arguments here follow the conventions required by `solve_ivp`, but the actual solver used is up to implementer. :param compute_dy_dt: Return ``dy_dt = compute_dy_dt(t, y)``. :type compute_dy_dt: function :param y0: Initial state. :type y0: array-like :param dy0: Initial direction to store. If `None`, then this is chosen randomly. :type dy0: array-like, None :param t0: Initial time. :type t0: float :param dt: Evolve for this length of time when computing the exponent. If `None`, then a reasonable value should be estimated by the code. (The code may adaptively update `dt`.) :type dt: float, None :param min_norm: Minimum norm. Start with states separated by `min_norm`, then evolve by `dt`, extract the exponent, add this to the samples, then pull the state back along the same direction to have length `min_norm` and repeat. If `None`, then reasonable values should be estimated by the code. :type min_norm: float, None :param Nsamples: Number of samples to use when estimating the Lyapunov exponent. The estimate should be the mean of this many samples with an error as the standard deviation. :type Nsamples: int :param norm: Use this function to compute the norm of the difference between states. (Default is `np.linalg.norm`.) :type norm: function :param rng: Random number generator such as returned by `np.random.default_rng()`, which is used by default if one is not provided. :type rng: random number generator :param debug: If `True`, then return `(lams, ts, ys, dys)` with the sample evolution. :type debug: bool :param solve_ivp_args: Additional arguments for `solve_ivp`. :type solve_ivp_args: dict, None :returns: * **lams** (*array of floats*) -- Array of maximal Lyapunov exponents such that the mean and standard deviations give a good estimate. These should be uncorrelated. * **ts, ys, dys** (*array*) -- Only provided if `debug` is `True`. Times, states, and separations used in sampling.