arXiv subjects

Khosro Tajbakhsh

Publications and source records attributed to Khosro Tajbakhsh.

7 recordsLinked to original sources

arXiv2026

Countable IET Models and Defect Sets for Interval Translation Maps

Interval translation maps are piecewise translations for which the images of distinct continuity intervals may overlap. We study when their measured dynamics can be represented by finite or countable interval exchange transformations. First, we give a direct entropy-based proof of the known fact that an interval translation map is invertible almost everywhere with respect to every nonatomic invariant probability measure. The proof uses a polynomial upper bound for the complexity of the natural branch coding. We then use a consequence of a theorem of Arnoux, Ornstein, and Weiss: every nonatomic measure-preserving automorphism of a standard probability space admits a countable interval exchange transformation model. Applied to the almost-everywhere invertible core, this gives an abstract countable interval exchange transformation model for every measured interval translation map. We next study the canonical order-preserving coordinate defined by the distribution function of the invariant measure. For each branch, we introduce a positive defect measure recording the excess measure carried by its direct image. Its push-forward to the distribution coordinate gives a canonical cut set such that, away from this set, the induced map is locally a translation. Finite defect support yields a finite interval exchange transformation model, while a Lebesgue-null defect cut set yields a countable model. Under branchwise nonsingularity, the defect support is the closure of the cuts arising from active gaps in the support, giving an equivalent geometric characterization of the finite case. Finally, for a self-similar infinite-type Bruin-Troubetzkoy map, we compute the canonical defect cut set explicitly. It is countably infinite and Lebesgue-null, yielding a genuinely countable, non-finite interval exchange transformation model.

math.DS↗

arXiv2025

Residual fertility and delay in sterile insect population dynamics

The sterile insect technique controls mosquito-borne diseases such as malaria, dengue, and yellow fever through either eradication or depressing the associated vector population. We formulate a three-dimensional delayed mosquito population suppression model with a saturated release rate to explore the interactive dynamics between wild, sterile, and non-sterile mosquitoes, focusing on the delay and residual fertility in the interactive dynamics among insects. We investigate the stability of the positive equilibrium and derive the Hopf bifurcation conditions. We establish the stability conditions for the positive equilibrium and examine how the time delay ($τ$) and residual fertility affect the non-sterile insects' dynamics. Below the critical values of the delay, the system remains stable, while beyond that, the Hopf bifurcation is guaranteed under certain circumstances. However, analysis shows a clear band of non-sterile insect population values as residual fertility varies within a very narrow range. This suggests that within this interval, the system exhibits sensitive dependence on the fertility parameter, likely due to underlying nonlinear dynamics. Numerical simulations are presented to support our analytical results, followed by a brief discussion of the findings.

math.DS↗

arXiv2023

Dynamics of Performances in a Competitive Model

A competitive resource-consumer dynamical model is analyzed based on an integrated model of a competitive Lotka-Volterra model and a prey-predator Rosenzweig-MacArthur model that we call that LV-RM model throughout this paper. Resource growth in the absence of consumers is logistic, and competing consumers' type II Holling's functional response made the model structure more realistic. We used the normal form and the center manifold theorems for bifurcation analysis of the presented model, identified Hopf and zero-Hopf bifurcations and their directions, and discussed their biological interpretations. We hypothesized that differentiated time scales of the competing consumers' predatory that lead to asymmetry in competition are the mechanisms that promote coexistence through relaxation-oscillation dynamics. Though, other performance parameters of both competitors are the same. Graphical representation of variations of the first Lyapunov coefficient, after competition coefficients interplay, shows various dynamics with growing complexity from the periodic state towards chaotic motion like Rössler attractor. We presented simulations to visualize the theoretical results obtained through bifurcation analysis.

math.DS↗

arXiv2021

Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear~naturally.

math.DS↗

arXiv2018

Specification properties on uniform spaces

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space $X$ with at least two elements, a nonempty set $Γ$ and a self--map $φ:Γ\toΓ$ the generalized shift dynamical system $(X^Γ,σ_φ)$: \begin{itemize} \item has (almost) weak specification property if and only if $φ:Γ\toΓ$ does not have any periodic point, \item has (uniform) stroboscopical property if and only if $φ:Γ\toΓ$ is one-to-one. \end{itemize}

math.DS↗

arXiv2018

Classification of special Anosov endomorphisms of nil-manifolds

In this paper we give a classification of special endomorphisms of nil-manifolds: Let $f:N/Γ\rightarrow N/Γ$ be a covering map of a nil-manifold and denote by $A:N/Γ\rightarrow N/Γ$ the nil-endomorphism which is homotopic to $f$. If $f$ is a special $TA$-map, then $A$ is a hyperbolic nil-endomorphism and $f$ is topologically conjugate to $A$.

math.DS↗

arXiv2017

Topology of pre-images under Anosov endomorphisms

For an endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. The inverse has been shown for a residual set of points but the the exact inverse has not yet been investigated before. Here we are going to show that under some conditions it is true for Anosov endomorphisms on closed manifolds, by using the fact that Anosov endomorphisms are covering maps.

math.DS↗