302 research outputs found
Resonance and marginal instability of switching systems
We analyse the so-called Marginal Instability of linear switching systems,
both in continuous and discrete time. This is a phenomenon of unboundedness of
trajectories when the Lyapunov exponent is zero. We disprove two recent
conjectures of Chitour, Mason, and Sigalotti (2012) stating that for generic
systems, the resonance is sufficient for marginal instability and for
polynomial growth of the trajectories. We provide a characterization of
marginal instability under some mild assumptions on the sys- tem. These
assumptions can be verified algorithmically and are believed to be generic.
Finally, we analyze possible types of fastest asymptotic growth of
trajectories. An example of a pair of matrices with sublinear growth is given
Qualitative stability patterns for Lotka-Volterra systems on rectangles
We present a qualitative analysis of the Lotka-Volterra differential equation within rectangles that are transverse with respect to the flow. In similar way to existing works on affine systems (and positively invariant rectangles), we consider here nonlinear Lotka-Volterra n-dimensional equation, in rectangles with any kind of tranverse patterns. We give necessary and sufficient conditions for the existence of symmetrically transverse rectangles (containing the positive equilibrium), giving notably the method to build such rectangles. We also analyse the stability of the equilibrium thanks to this transverse pattern. We finally propose an analysis of the dynamical behavior inside a rectangle containing the positive equilibrium, based on Lyapunov stability theory. More particularly, we make use of Lyapunov-like functions, built upon vector norms. This work is a first step towards a qualitative abstraction and simulation of Lotka-Volterra systems
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