arXiv subjects

Hassan Najafi Alishah

Publications and source records attributed to Hassan Najafi Alishah.

8 recordsLinked to original sources

arXiv2023

The Hamiltonian formalism of the Inverse Problem

A big-isotropic structure is a generalization of the notion of Dirac structure, due to Vaisman. We discuss the inverse problem of deciding if a vector field is Hamiltonian having a big-isotropic structure as underlying geometry. In [1] we have considered this question for the special case of replicator equations. Here we generalize that approach to any vector field that can be written in the form $X(u)=(\Bη)(u),$ where $\B=\B(u)$ is a matrix and $η=η(u)$ is a vector. For a linear system we show that, if the representing matrix of the system has at least one pair of positive-negative non-zero eigenvalues, or a zero eigenvalue with at least one 3-dimensional Jordan block associated to it, then the linear system has a Hamiltonian description with respect to a big-isotropic structure. As a byproduct, we find a class of linear systems with zero eigenvalue that are Hamiltonian with respect to a big-isotropic structure but not a Dirac structure. Moreover, we prove that every linear Hamiltonian system, having a big-isotropic structure as underlying geometry, is completely integrable in the sense of Zung [12]. For linear systems, in the both Hamiltonian formulation and complete integrability cases, explicit descriptions of the geometric structures, the Hamiltonian functions, the first integrals and the commuting flows are provided.

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arXiv2021

Asymptotic Dynamics of Hamiltonian Polymatrix Replicators

In a previous paper [3] we have studied flows defined on polytopes, presenting a new method to encapsulate its asymptotic dynamics along the edge-vertex heteroclinic network. These results apply to the class of polymatrix replicator systems, which contains several important models in Evolutionary Game Theory. Here we establish the Hamiltonian character of the asymptotic dynamics of Hamiltonian polymatrix replicators.

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arXiv2019

Conservative Replicator and Lotka-Volterra Equations in the context of Dirac$\backslash$ big-isotropic Structures

We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a time re-parametrization, is provided using Dirac$\backslash$big-isotropic structures. Using the equivalence between replicator and Lotka-Volterra (LV) equations, the set of conservative LV equations is enlarged. Our approach generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system. In the case of predator-prey model, our method does allow interaction between different predators and between different preys.

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arXiv2018

Asymptotic Poincaré Maps along the edges of Polytopes

For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope's vertexes and edges. This piecewise linear flow is easy to compute even in higher dimensions, which allows the usage of numeric algorithms to find invariant dynamical structures such as periodic, homoclinic or heteroclinic orbits, which if robust persist as invariant dynamical structures of the original flow. We apply this method to prove the existence of chaotic behavior in some Hamiltonian replicator systems on the five dimensional simplex.

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arXiv2015

Conservative and Dissipative Polymatrix Replicators

In this paper we address a class of replicator dynamics, referred as polymatrix replicators, that contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for $n$-person games. Polymatrix replicators form a simple class of algebraic o.d.e.'s on prisms (products of simplexes), which describe the evolution of strategical behaviours within a po\-pu\-lation stratified in $n\geq 1$ social groups. In the 80's Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems. This theory is built around a reduction algorithm that "infers" the localization of the system' s attractor in some affine subspace. It was later proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system. In this paper we extend these results to polymatrix replicators.

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arXiv2014

Hamiltonian Evolutionary Games

We introduce a class of o.d.e.'s that generalizes to polymatrix games the replicator equations on symmetric and asymmetric games. We also introduce a new class of Poisson structures on the phase space of these systems, and characterize the corresponding subclass of Hamiltonian polymatrix replicator systems. This extends known results for symmetric and asymmetric replicator systems.

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arXiv2013

Realization of tangent perturbations in discrete and continuous time conservative systems

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.

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arXiv2012

Tracing KAM tori in presymplectic dynamical systems

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $ω$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $ω$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.

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