Semiclassical methods can now explain many mesoscopic effects (shot-noise, conductance fluctuations, etc) in clean chaotic systems, such as chaotic quantum dots. In the deep classical limit (wavelength much less than system size) the Ehrenfest time (the time for a wavepacket to spread to a classical size) plays a crucial role, and random matrix theory (RMT) ceases to apply to the transport properties of open chaotic systems. Here we summarize some of our recent results for shot-noise (intrinsically quantum noise in the current through the system) in this deep classical limit. For systems with perfect coupling to the leads, we use a phase-space basis on the leads to show that the transmission eigenvalues are all 0 or 1 -- so transmission is noiseless [Whitney-Jacquod, Phys. Rev. Lett. 94, 116801 (2005), Jacquod-Whitney, Phys. Rev. B 73, 195115 (2006)]. For systems with tunnel-barriers on the leads we use trajectory-based semiclassics to extract universal (but non-RMT) shot-noise results for the classical regime [Whitney, Phys. Rev. B 75, 235404 (2007)].
Two symmetrically coupled populations of N oscillators with inertia $m$ display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendula. In particular, we report the first evidence of intermittent chaotic chimeras, where one population is synchronized and the other jumps erratically between laminar and turbulent phases. These states have finite life-times diverging as a power-law with N and m. Lyapunov analyses reveal chaotic properties in quantitative agreement with theoretical predictions for globally coupled dissipative systems.
Billiard systems offer a simple setting to study regular and chaotic dynamics. Gravitational billiards are generalizations of these classical billiards which are amenable to both analytical and experimental investigations. Most previous work on gravitational billiards has been concerned with two dimensional boundaries. In particular the case of linear boundaries, also known as the wedge billiard, has been widely studied. In this work, we introduce a three dimensional version of the wedge; that is, we study the nonlinear dynamics of a billiard in a constant gravitational field colliding elastically with a linear cone of half angle $theta$. We derive a two-dimensional Poincar{e} map with two parameters, the half angle of the cone and $ell$, the $z$-component of the billiards angular momentum. Although this map is sufficient to determine the future motion of the billiard, the three-dimensional nature of the physical trajectory means that a periodic orbit of the mapping does not always correspond to a periodic trajectory in coordinate space. We demonstrate several integrable cases of the parameter values, and analytically compute the systems fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of $ell$ the conic billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase $ell$ the dynamics becomes on the whole less chaotic, and the correspondence with the wedge billiard is lost.
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.
Geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical- and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme.
An optimization based state and parameter estimation method is presented where the required Jacobian matrix of the cost function is computed via automatic differentiation. Automatic differentiation evaluates the programming code of the cost function and provides exact values of the derivatives. In contrast to numerical differentiation it is not suffering from approximation errors and compared to symbolic differentiation it is more convenient to use, because no closed analytic expressions are required. Furthermore, we demonstrate how to generalize the parameter estimation scheme to delay differential equations, where estimating the delay time requires attention.
Features of the Jacobian matrix of the delay coordinates map are exploited for quantifying the robustness and reliability of state and parameter estimations for a given dynamical model using an observed time series. Relevant concepts of this approach are introduced and illustrated for discrete and continuous time systems employing a filtered Henon map and a Rossler system.
Observability of state variables and parameters of a dynamical system from an observed time series is analyzed and quantified by means of the Jacobian matrix of the delay coordinates map. For each state variable and each parameter to be estimated a measure of uncertainty is introduced depending on the current state and parameter values, which allows us to identify regions in state and parameter space where the specific unknown quantity can (not) be estimated from a given time series. The method is demonstrated using the Ikeda map and the Hindmarsh-Rose model.
Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment).
The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulae for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.
