Dynamical Systems Theory Compared to Game Theory: The Case of the Salamis’s Battle

In this paper, we present an innovative non–linear, discrete, dynamical system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks the anniversary of the 2500 years that have passed since this famous naval battle which took place in late September 480 B.C. The suggested model describes very well the most effective strategic behavior between two participants during a battle (or in a war). Moreover, we compare the results of the Dynamical Systems analysis to Game Theory, considering this conflict as a “war game”.

Modeling and Stability Analysis of a Communication Network System

In this work, the modeling and stability problem for a communication network system is addressed. The communication network system consists of a transmitter which sends messages to a receiver. The proposed model considers two possibilities. The first one, that messages are successfully received, while in the second one, during the sending process the transmitter breaks down and as a result the message does not reach the receiver. Timed Petrinets is the mathematical and graphical modeling technique utilized. Lyapunov stability theory provides the required tools needed to aboard the stability problem. Employing Lyapunov methods, a sufficient condition for stabilization is obtained. It is shown that it is possible to restrict the communication network system state space in such a way that boundedness is guaranteed. However, this restriction results to be vague. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.

Compatible-invariant subset analysis of deterministic finite automata via semi-tensor product of matrices approach

The compatible-invariant subset of deterministic finite automata（ DFA） is investigated to solve the problem of subset stabilization under the frameworks of semi-tensor product（ STP） of matrices. The concepts of compatibleinvariant subset and largest compatible-invariant subset are introduced inductively for Moore-type DFA,and a necessary condition for the existence of largest compatible-invariant subset is given. Meanwhile,by using the STP of matrices,a compatible feasible event matrix is defined with respect to the largest compatible-invariant subset.Based on the concept of compatible feasible event matrix,an algorithm to calculate the largest compatible-invariant subset contained in a given subset is proposed. Finally,an illustrative example is given to validate the results.

Complexity of comparing expressions in max-plus algebra

Max-plus algebra has been widely used in the study of discrete-event dynamic systems.Using max-plus algebra makes it easy to specify safety constraints on events since they can be described as a set of inequalities of state variables,i.e.,firing times of relevant events.This paper proves that the problem of solving max-plus inequalities in a cube(MAXINEQ)is nondeterministic polynomial-time hard(NP-hard)in strong sense and the problem of verifying max-plus inequalities(VERMAXINEQ)is co-NP.As a corollary,the problem of solving a system of multivariate max-algebraic polynomial equalities and inequalities(MPEI)is shown to be NP-hard in strong sense.The results indicate the difficulties in comparing max-plus formulas in general.Problem structures of specific systems have to be explored to enable the development of efficient algorithms.