(c) Examples of exotic looped trajectories arising from the higher order terms in equation (8).
Due to the discontinuous trajectories arising from the nonsmooth impacts, we use hybrid systems stability analysis tools to establish that 1) a tracking control algorithm and 2) an observer algorithm guarantee global exponential stability.
That study also found that two slow variables are necessary to construct the folded bifurcation plane on which these complicated bursting trajectories arise.
Chaos is characterised roughly as extreme sensitivity in the behaviour of a system on its initial conditions, in the sense that the distance between the trajectories arising from different initial conditions increases exponentially in time.
The comparison between the HDE trajectories arising from different input histories quantifies the intrinsic discrimination capabilities of single neurons, and their dependence upon the input statistics and the neuronal dynamics.
Six of the seven origins of the free-living state were parallel trajectories arising from byssal attaching ancestors.
Here, by constructing a two-dimensional energy landscape from the individual landscapes of active and backtracked enzymes, we identify a new pushing mechanism which is the result of a saddle trajectory that arises in the two-dimensional energy landscape of interacting enzymes.
Pinch branches and bundles of trajectories can arise only when certain relationships of K are satisfied on edges of the concentration simplex.
Ensemble averaging trajectories that arise from different mechanisms, e.g., actively driven and diffusive, is incorrect and can result inaccurate correlations between structure, mechanics, and activity.
Ancestral sequence inference, directed evolution, structural analysis, NMR, and molecular dynamics simulations illuminate how enantioselective activity arises during the evolutionary trajectory of chalcone isomerase from a noncatalytic ancestor.
