# American Institute of Mathematical Sciences

ISSN:
2158-2491

eISSN:
2158-2505

All Issues

## Journal of Computational Dynamics

June 2014 , Volume 1 , Issue 2

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2014, 1(2): 213-232 doi: 10.3934/jcd.2014.1.213 +[Abstract](3423) +[PDF](916.4KB)
Abstract:
We consider the global asymptotic stability of the trivial fixed point of the difference equation $x_{n+1}=m x_n-\alpha \varphi(x_{n-1})$, where $(\alpha,m) \in \mathbb{R}^2$ and $\varphi$ is a real function satisfying the discrete Yorke condition: $\min\{0,x\} \leq \varphi(x) \leq \max\{0,x\}$ for all $x\in \mathbb{R}$. If $\varphi$ is bounded then $(\alpha,m) \in [|m|-1,1] \times [-1,1]$, $(\alpha,m) \neq (0,-1), (0,1)$ is necessary for the global stability of $0$. We prove that if $\varphi(x) \equiv \tanh(x)$, then this condition is sufficient as well.
2014, 1(2): 233-248 doi: 10.3934/jcd.2014.1.233 +[Abstract](2277) +[PDF](432.0KB)
Abstract:
From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.
2014, 1(2): 249-278 doi: 10.3934/jcd.2014.1.249 +[Abstract](3350) +[PDF](8600.4KB)
Abstract:
The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the efficacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.
2014, 1(2): 279-306 doi: 10.3934/jcd.2014.1.279 +[Abstract](3538) +[PDF](1246.3KB)
Abstract:
We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy first reduce, then optimize''---rather than first optimize, then reduce''---is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the first reduce, then control'' strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the first reduce, then optmize'' approach and discuss possible pitfalls.
2014, 1(2): 307-338 doi: 10.3934/jcd.2014.1.307 +[Abstract](2117) +[PDF](584.3KB)
Abstract:
We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.
2014, 1(2): 339-356 doi: 10.3934/jcd.2014.1.339 +[Abstract](2141) +[PDF](982.3KB)
Abstract:
We propose a method for approximating solutions to optimization problems involving the global stability properties of parameter-dependent continuous-time autonomous dynamical systems. The method relies on an approximation of the infinite-state deterministic system by a finite-state non-deterministic one --- a Markov jump process. The key properties of the method are that it does not use any trajectory simulation, and that the parameters and objective function are in a simple (and except for a system of linear equations) explicit relationship.
2014, 1(2): 357-375 doi: 10.3934/jcd.2014.1.357 +[Abstract](2612) +[PDF](951.9KB)
Abstract:
Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.
2014, 1(2): 377-389 doi: 10.3934/jcd.2014.1.377 +[Abstract](2029) +[PDF](551.1KB)
Abstract:
An algorithm, based on the Equation-free concept, for the approximation of coarse-grained center manifolds of microscopic simulators is addressed. It is assumed that the macroscopic equations describing the emergent dynamics are not available in a closed form. Appropriately initialized short runs of the microscopic simulators, which are treated as black box input-output maps provide a polynomial estimate of a local coarse-grained center manifold; the coefficients of the polynomial are obtained by wrapping around the microscopic simulator an optimization algorithm. The proposed method is demonstrated through kinetic Monte Carlo simulations, of simple reactions taking place on catalytic surfaces, exhibiting coarse-grained turning points and Andronov-Hopf bifurcations.
2014, 1(2): 391-421 doi: 10.3934/jcd.2014.1.391 +[Abstract](14982) +[PDF](1657.5KB)
Abstract:
Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.