Predicting chaotic statistics with unstable invariant tori.

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent sums over periodic orbits, which are inspired by the rigorous periodic orbit theory and which have shown much promise in fluid dynamics. Using a new numerical method for converging unstable invariant two-tori in a chaotic partial differential equation (PDE), and exploiting symmetry breaking of relative periodic orbits to detect those tori, we identify many quasiperiodic, unstable, invariant two-torus solutions of a modified Kuramoto-Sivashinsky equation. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori-with weights computed based on their respective stability eigenvalues-approximate average quantities for the chaotic dynamics. These results are a step toward exploiting higher-dimensional invariant sets to describe general hyperchaotic systems, including dissipative spatiotemporally chaotic PDEs.

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems.

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.

Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method.

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatiotemporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification, or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent, and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatiotemporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

Right and left ventricular function in hospitalized children with respiratory syncytial virus infection.

BACKGROUND: Extrapulmonary manifestations including cardiac dysfunction have been demonstrated in children with respiratory syncytial virus (RSV) infection requiring intensive care. The aim of this study was to examine cardiac function in hospitalized children with moderate RSV infection admitted to a regular pediatric ward. METHODS: We used echocardiography to determine cardiac output, and right and left ventricular function in 26 patients (aged 2 weeks to 24 months) with RSV infection. The echocardiographic results were compared with s-troponin, the need for supplementary oxygen or noninvasive respiratory support, and capillary refill time. RESULTS: The number of measured s-troponins (ten [38%] of the included children) was too low to assess differences between children with elevated levels and those with normal levels. There were no differences in cardiac function between patients receiving oxygen treatment or respiratory support and those who did not. Capillary refill time did not correlate with any of the echocardiographic parameters. Both left and right ventricular output (mL/kg/min) was higher than published reference values. All other echocardiographic parameters were within the reference range. CONCLUSION: Children with moderate RSV infection had an increased left and right ventricular output, and cardiac function was well maintained. We conclude that routine cardiac ultrasound is not warranted in children with moderate RSV infection. The role of an elevated s-troponin in these patients remains to be determined.