Ant algorithms are one of the main programming paradigms in swarm intelligence. They are built on stochastic decision functions, which can also be found in other types of bio-inspired algorithms with the same mathematical form. However, though this modeling leads to high-performance algorithms, some phenomena, like symmetry break, are still not well understood or modeled at the ant level. This paper proposes an original analysis of the problem : we establish a reactive multi-agent system based on logistic nonlinear decision maps, and designed according to the influence-reaction scheme. Our proposition is an entirely novel approach to the mathematical foundations of ant algorithms : contrary to the current stochastic approaches, we show that an alternative deterministic model exists, which has its origin in deterministic chaos theory. The rewriting of the decision functions leads to a new way of understanding and visualizing the convergence behavior of ant algorithms. We apply our approach on a concrete example, namely the binary bridge problem.