The developmental dynamics of multicellular organisms is a process that takes place in a multistable system in which each attractor state represents a cell type, and attractor transitions correspond to cell differentiation paths. This new understanding has revived the idea of a quasipotential landscape, first proposed by Waddington as a metaphor. To describe development, one is interested in the 'relative stabilities' of N attractors (N . 2). Existing theories of state transition between local minima on some potential landscape deal with the exit part in the transition between two attractors in pair-attractor systems but do not offer the notion of a global potential function that relates more than two attractors to each other. Several ad hoc methods have been used in systems biology to compute a landscape in non-gradient systems, such as gene regulatory networks. Here we present an overview of currently available methods, discuss their limitations and propose a new decomposition of vector fields that permits the computation of a quasi-potential function that is equivalent to the Freidlin-Wentzell potential but is not limited to two attractors. Several examples of decomposition are given, and the significance of such a quasi-potential function is discussed.

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SUMMARYIn this paper, we consider the mixed H 2 /H ∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of this problem with a finite-dimensional filter are given in terms of a pair of coupled Hamilton-Jacobi-Isaacs equations (HJIEs). For linear systems, it is shown that these conditions reduce to a pair of coupled Riccati equations similar to the ones for the control case. Both the finite-horizon and the infinite-horizon problems are discussed. Simulation results are presented to show the usefulness of the scheme, and the results are generalized to include other classes of nonlinear systems.

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In this paper, we consider the ℋ︁∞‐filtering problem for singularly perturbed (two time‐scale) nonlinear systems. Two types of filters are discussed, namely, (i) decomposition and (ii) aggregate, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Isaac's equations (HJIEs) are presented. Reduced‐order filters are also derived in each case, and the results are specialized to linear systems, in which case the HJIEs reduce to a system of linear‐matrix‐inequalities (LMIs). Based on the linearization of the nonlinear models, upper bounds ε* of the singular parameter ε that guarantee the asymptotic stability of the nonlinear filters can also be obtained. The mixed ℋ︁2/ℋ︁∞‐filtering problem is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.