arXiv subjects

Nataliya A. Balabanova

Publications and source records attributed to Nataliya A. Balabanova.

9 recordsLinked to original sources

arXiv2026

Adaptive dynamics of alternating Prisoner's Dilemma with memory N

The Prisoner's Dilemma is used as a model in processes involving reciprocity; however, its classical setup can be insufficient in settings where the symmetry of the simultaneous decision making is broken -- for example, in donor and recipient processes. In the alternating Prisoner's Dilemma model the two players take turns choosing their strategy. Assuming a finite memory setup, we establish the mathematical aspects of the adaptive dynamics of the alternating Prisoner's Dilemma, paying particular attention to the case of memory 1.

math.DS↗

arXiv2025

A Hamiltonian approach for point vortices on non-orientable surfaces

We investigate the motion of point vortices on the Mobius band and Klein bottle. Since these are non-orientable surfaces, the standard Hamiltonian approach does not apply. We therefore begin by establishing a modified Hamiltonian approach which works for arbitrary non-orientable surfaces, through describing the phase space, the Hamiltonian and the local equations of motion. We use a combination of twisted functions and oriented double covers to adapt some of the known notions of vortex dynamics to non-orientable surfaces. For both of the surfaces of interest, we write Hamiltonian-type equations of vortex motion explicitly and follow that by the description of relative equilibria and an investigation of the motion of one and two vortices.

math.DS↗

arXiv2025

Replicator-mutator dynamics for public goods games with institutional incentives

Understanding the emergence and stability of cooperation in public goods games is important due to its applications in fields such as biology, economics, and social science. However, a gap remains in comprehending how mutations, both additive and multiplicative, as well as institutional incentives, influence these dynamics. In this paper, we study the replicator-mutator dynamics, with combined additive and multiplicative mutations, for public goods games both in the absence or presence of institutional incentives. For each model, we identify the possible number of (stable) equilibria, demonstrate their attainability, as well as analyse their stability properties. We also characterise the dependence of these equilibria on the model's parameters via bifurcation analysis and asymptotic behaviour. Our results offer rigorous and quantitative insights into the role of institutional incentives and the effect of combined additive and multiplicative mutations on the evolution of cooperation in the context of public goods games.

math.DS↗

arXiv2024

Adaptive dynamics for individual payoff game-theoretic models of vaccination

Vaccination is widely recognised as one of the most effective forms of public health interventions. Individuals decisions regarding vaccination creates a complex social dilemma between individual and collective interests, where each person's decision affects the overall public health outcome. In this paper, we study the adaptive dynamics for the evolutionary dynamics of strategies in a fundamental game-theoretic model of vaticination. We show the existence of an (Nash) equilibrium and analyse the stability and bifurcations when varying the relevant parameters. We also demonstrate our analytical results by several concrete examples.

math.DS↗

arXiv2024

Adaptive dynamics of direct reciprocity with N rounds of memory

The theory of direct reciprocity explores how individuals cooperate when they interact repeatedly. In repeated interactions, individuals can condition their behaviour on what happened earlier. One prominent example of a conditional strategy is Tit-for-Tat, which prescribes to cooperate if and only if the co-player did so in the previous round. The evolutionary dynamics among such memory-1 strategies have been explored in quite some detail. However, obtaining analytical results on the dynamics of higher memory strategies becomes increasingly difficult, due to the rapidly growing size of the strategy space. Here, we derive such results for the adaptive dynamics in the donation game. In particular, we prove that for every orbit forward in time, there is an associated orbit backward in time that also solves the differential equation. Moreover, we analyse the dynamics by separating payoffs into a symmetric and an anti-symmetric part and demonstrate some properties of the anti-symmetric part. These results highlight some interesting symmetries that arise when interchanging player one with player two, and cooperation with defection.

math.DS↗

arXiv2023

A note on the geometry of the two-body problem on $S^2$

Leveraging on the results of arXiv:2210.13644 , we carry out an investigation of the algebraic three-fold $Σ_{C,h}$, the common level set of the Hamiltonian and the Casimir, for the two-body problem for equal masses on $S^2$ subject to a gravitational potential of cotangent type. We determine the topology of its compactification $\overlineΣ_{C,h}$ and how it bifurcates with respect to the admissible values of $(C,h)$, ($C$ being the fixed value of the Casimir and $h$ the fixed value of the Hamiltonian). This bifurcation diagram is actually equal to the bifurcation diagram that describes relative equilibria. We also prove that for $h$ sufficiently negative $Σ_{C,h}$ is equipped with a global contact form obtained from the environment symplectic form via a suitable Liouville vector field.

math.DS↗

arXiv2023

Collision trajectories and regularisation of two-body problem on $S^2$

In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position vectors of the two bodies. We describe the $ω$-limit set of the variables in the symplectically reduced system corresponding to initial data that lead to collisions. Furthermore we provide a geometric description of the dynamics. Lastly, we regularise the system and investigate its behaviour on near collision orbits. This involves the study of completely degenerate equilibria and the use of high-dimensional non-homogenous blow-ups.

math.DS↗

arXiv2022

A Hamiltonian approach for point vortices on non-orientable surfaces II: the Klein bottle

This is the second of two companion papers dedicated to the investigation of vortex motion on non-orientable surfaces. The first paper of the pair is predominantly concerned with establishing the Hamiltonian approach to systems of point vortices on non-orientable manifolds and investigating the limits of the intrinsic (restricted to the non-orientable manifold) approach. In addition, point vortex motion on the Mobius band is closely examined. In this paper, we investigate dynamics of one and two point vortices on the Klein bottle through establishing explicit forms of the Hamiltonian, equations of motion on the charts, describing relative equilibria, etc.

math.DS↗

arXiv2021

Two body problem on a sphere in the presence of a uniform magnetic field

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles, interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.

math-ph↗