This paper examines and contrasts the feasibility of joint state and parameter estimation of noisedriven chaotic systems using the extended Kalman filter (EKF), ensemble Kalman filter (EnKF), and particle filter (PF). In particular, we consider the chaotic vibration of a noisy Duffing oscillator perturbed by combined harmonic and random inputs ensuing a transition probability density function (pdf) of motion which displays strongly non-Gaussian features. This system offers computational simplicity while exhibiting a kaleidoscope of dynamical behavior with a slight change of input and system parameters. An extensive numerical study is undertaken to contrast the performance of various nonlinear filtering algorithms with respect to sparsity of observational data and strength of model and measurement noise. In general, the performance of EnKF is better than PF for smaller ensemble size, while for larger ensembles PF outperforms EnKF. For moderate measurement noise and frequent measurement data, EKF is able to correctly track the dynamics of the system. However, EKF performance is unsatisfactory in the presence of sparse observational data or strong measurement noise.

A central issue in complex networks is tolerance of random failures and intentional attacks. Current literature emphasizes the dichotomy between networks with a power-law node connectivity distribution, which are robust to random failures but fragile to targeted attacks, versus networks with an exponentially decaying connectivity distribution, which are less tolerant to failures but more resilient to attacks. We prove analytically that the optimal network configuration under a classic measure of robustness is altogether different from both of the above: in all cases, failure and/or attack, there are no more than three distinct node connectivities in the optimal network.

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