arXiv subjects

Michael Zaks

Publications and source records attributed to Michael Zaks.

3 recordsLinked to original sources

arXiv2026

Seasonal Forcing in Rock-Paper-Scissors Population Dynamics

We study a class of cyclic dominance models with seasonal forcing, extending the classical May-Leonard competition framework. By introducing time-periodic coefficients into the growth rates and the interaction terms, we explore how environmental seasonality influences the dynamics of three-species systems. Through analytical estimates and numerical simulations, we reveal the emergence of complex oscillatory behavior, including multi-year periodic cycles, transitions to heteroclinic cycles, and chaotic oscillations. Our results highlight how periodic modulation can destabilize stable periodic trajectories, generate novel attractors, and give rise to coexistence mechanisms not present in autonomous systems. These findings contribute to the understanding of biodiversity maintenance under realistic, temporally varying ecological conditions.

math.DS↗

arXiv2021

Basin sizes depend on stable eigenvalues in the Kuramoto model

We show that for the Kuramoto model (with identical phase oscillators equally coupled) its global statistics and size of the basins of attraction can be estimated through the eigenvalues of all stable (frequency) synchronized states. This result is somehow unexpected since, by doing that, one could just use local analysis to obtain global dynamic properties. But recent works based on Koopman and Perron-Frobenius operators demonstrate that global features of a nonlinear dynamical system, with some specific conditions, are somehow encoded in the local eigenvalues of its equilibrium states. Recognized numerical simulations in the literature reinforce our analytical results.

nlin.CD↗

arXiv2010

Levels in the toposes of simplicial sets and cubical sets

The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of n-skeletal and n-coskeletal simplicial sets. In addition to the obvious ordering, the levels provide a stricter means of comparing the complexity of objects, which is determined by the answer to the following question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This paper answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical sets, and reflexive globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the other two examples the situation is considerably more complicated: n-skeletal implies (2n-1)-coskeletal for simplicial sets and 2n-coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.

math.CT↗