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A necessary condition for the emergence of chaos is given. It is well known that the emergence of chaos requires a positive exponent which entails diverging trajectories. Here we show that this is not enough. An additional necessary condition for the emergence of chaos in the region where the trajectory of the system goes through, is that the product of the maximal positive exponent times the duration in which the system configuration point stays in the unstable region should exceed unity. We give a theoretical analysis justifying this result and a few examples.
We consider a $N$-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit $N to infty$, as following from the previous analysis in
We show that the dissipation rate bounds the rate at which physical processes can be performed in stochastic systems far from equilibrium. Namely, for rare processes we prove the fundamental tradeoff $langle dot S_text{e} rangle mathcal{T} geq k_{tex
We study the spectral statistics of the Dirac operator on a rose-shaped graph---a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac
This article treats chaotic scattering with three degrees of freedom, where one of them is open and the other two are closed, as a first step toward a more general understanding of chaotic scattering in higher dimensions. Despite of the strong rest
We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we establish a local condition of second derivative type under which the crit