arXiv subjects

Nasrin Sadri

Publications and source records attributed to Nasrin Sadri.

6 recordsLinked to original sources

arXiv2025

FitzHugh-Nagumo equation: bifurcations, slow-fast system and dynamics near infinity

In this paper, we present a qualitative and bifurcation analysis of the three-parameter FitzHugh-Nagumo system and its compactified formulation. The study is structured according to three parameter-dependent regimes, for which the associated phase portraits are characterized. In one of these regimes, the system exhibits a double-zero bifurcation with $Z_2$ symmetry, corresponding to a codimension-two degeneracy. We compute explicit bifurcation and transition curves arising in the unfolding of this singularity, including pitchfork, Hopf, Belyakov, and double homoclinic bifurcations, and we construct the corresponding bifurcation diagrams. The local bifurcation analysis is linked to the slow-fast structure of the model, highlighting the emergence of canard solutions and their role in organizing the dynamics. In addition, we analyze the global behavior of the system by studying the dynamics at infinity through compactification techniques. The results obtained complement and extend earlier studies on global bifurcations in the FitzHugh-Nagumo model, in particular those reported by Georgescu, Rocsoreanu, and Giurgiteanu in "Global Bifurcations in FitzHugh-Nagumo Model", Trends in Mathematics: Bifurcations, Symmetry and Patterns (2003).

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arXiv2025

Global Centers and Phase Portraits in Generalized Duffing Oscillators: A Comprehensive Study of the Center-Focus Problem

This work presents a comprehensive study of the generalized Duffing oscillator, a fundamental model in nonlinear dynamics described by the system $$ \dot{x} = y, \quad \dot{y} = -\alpha y - \epsilon x^m - \sigma x, $$ where $\epsilon \neq 0$ and $m \geq 1$. We focus on the topological classification of phase portraits, the characterization of global centers, and the absence of limit cycles for $\alpha\neq0$. For the linear case ($m = 1$), we establish necessary and sufficient conditions for the origin to be a global center, showing that this occurs if, and only if, $\alpha = 0$ and $\epsilon + \sigma > 0$. For the nonlinear case ($m > 1$), we prove that the origin is a global center if, and only if, $m$ is odd, $\sigma, \epsilon > 0$, $\alpha = 0$. Additionally, we classify the global phase portraits for every $m$, demonstrating the rich dynamical behavior of the system and detect homoclinic, heteroclinic and double-homoclinic cycles for $\alpha=0$. Using the Bendixson-Dulac criterion, we rule out the existence of limit cycles for $\alpha\neq 0$, further clarifying the behavior of the system. Our results resolve the center-focus problem for the degenerate case $\alpha = 0$ and provide a complete characterization of global centers for generalized Duffing oscillators of odd degrees. These findings contribute to the broader understanding of nonlinear dynamical systems and have potential applications in modeling oscillatory phenomena.

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arXiv2024

Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System

The FitzHugh-Nagumo system is a $4$-parameter family of $3$D vector field used for modeling neural excitation and nerve impulse propagation. The origin represents a Hopf-zero equilibrium in the FitzHugh-Nagumo system for two classes of parameters. In this paper, we employ recent techniques in averaging theory to investigate, besides periodic solutions, the bifurcation of invariant tori within the aforementioned families. We provide explicit generic conditions for the existence of these tori and analyze their stability properties. Furthermore, we employ the backward differentiation formula to solve the stiff differential equations and provide numerical simulations for each of the mentioned results.

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arXiv2023

Symmetry-breaking singular controller design for Bogdanov-Takens bifurcations with an application to Chua system

We provide a complete symmetry-breaking bifurcation control for equivariant smooth differential systems with Bogdanov-Takens singularities. Controller coefficient space is partitioned by critical controller sets into different connected regions. The connected regions provide a classification for all qualitatively different dynamics of the controlled system. Hence, a state feedback controller design with four small controller coefficients is proposed for an efficient and full singular symmetry-breaking control. Our approach works well for nonlinear control systems with both controllable and uncontrollable linearizations. Origin is a primary equilibrium for the uncontrolled system. This gives rise to two secondary local equilibria for the controlled system. These equilibria further experience tertiary fold and hysteresis type bifurcations. The secondary and primary equilibria experience Hopf and Bautin bifurcations leading to the appearance of one limit cycle from primary equilibrium, and either one or two from each secondary equilibria. The collisions of limit cycles with equilibria lead to either a heteroclinic cycle or four different homoclinic cycles. Each pair of limit cycles may respectively merge together and disappear. This is a saddle-node bifurcation of limit cycles. Different combinations of these give rise to a rich list of bifurcation scenarios. Finite determinacy of each of these bifurcations has been thoroughly investigated. This greatly influences their stabilization potential in applications. We consider Chua system with a quadratic state-feedback controller. Controlled Chua system experiences a pitchfork bifurcation, three Hopf bifurcations and two homoclinic bifurcations. There exist two different regions of controller coefficient choices for feedback regularization and two nearby regions for supercritical Hopf stabilization approach.

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arXiv2017

Bifurcation controller designs for the generalized cusp plants of Bogdanov--Takens singularity with an application to ship control

Nonlinear controlled plants with Bogdanov-Takens singularity may experience surprising changes in their number of equilibria, limit cycles and/or their stability types when the controllers slightly vary in the vicinity of critical parameter varieties. Each such a change is called a local bifurcation. We derive novel results with regards to truncated parametric normal form classification of the generalized cusp plants. Then, we suggest effective nonlinear bifurcation control law designs for precisely locating and accurately controlling many different types of bifurcations for two measurable plants from this family. The first is a general quadratic plant with a possible multi-input linear controller while the second is a Z_2-equivariant general plant with possible multi-input linear (Z_2-symmetry preserving) and quadratic (symmetry-breaking) controllers. The bifurcations include from primary to quinary bifurcations of either of the following types: saddle-node, transcritical and pitchfork of equilibria, Z_2-equivariant bifurcations of multiple limit cycles through Hopf, homoclinic, heteroclinic, saddle-node, and saddle-connection, and finally their one-parameter symmetry breaking bifurcations. Using our parametric normal form analysis, we propose a new approach for efficient treatment of tracking and regulating engineering problems with smooth manoeuvering possiblities. Due to the nonlinearity of a ship maneuvering characteristic, there is a need for a controller design in a ship steering system so that the ship follows a desired sea route. The results in bifurcation control analysis are applied to two nonlinear ship course models for such controller designs. Symbolic implementations in Maple and numerical simulations in MATLAB confirm our theoretical results and accurate predictions.

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arXiv2016

Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms

In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locate local bifurcations in terms of these parameters. Here, we apply the proposed approach on Hopf-zero singularities whose the first few low degree terms are incompressible. In this direction, we obtain novel orbital and parametric normal form results for such families by assuming a nonzero quadratic condition. Moreover, we give a truncated universal asymptotic unfolding normal form and prove the finite determinacy of the steady-state bifurcations for two most generic subfamilies of the associated amplitude systems. We analyze the local bifurcations of equilibria, limit cycles and the secondary Hopf bifurcation of invariant tori. The results are successfully implemented and verified using Maple. By employing the proposed approach, we design an effective multiple-parametric quadratic state feedback controller for a singular system on a three dimensional central manifold with two imaginary uncontrollable modes. We illustrate how our program systematically identifies the distinguished (universal unfolding) parameters, derives the estimated transition varieties in terms of these parameters, and locates the local primary and secondary bifurcations of equilibria, limit cycles and invariant tori. This approach is useful in designing efficient nonlinear feedback controllers (single or multiple inputs) for local bifurcation control in engineering problems.

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