Mathematics Commons^™

On Local Asymptotic Stability Of Q-Fractional Nonlinear Dynamical Systems, Ilknur Koca, Elif Demirci

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, locally asymptotic stability of q-fractional order nonlinear dynamical systems is introduced and studied. The sufficient conditions for local stability of such dynamical systems are obtained. Also, useful definitions of fractional order q-integrals and q-derivatives are recalled. Finally, a q-fractional order nonlinear dynamical model is considered.

Applying Linear Controls To Chaotic Continuous Dynamical Systems, James P. Braselton, Yan Wu

Mathematical Sciences: Faculty Publications

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.

Stability And Data Dependence Results For The Jungck--Khan Iterative Scheme, Abdul Rahim Khan, Fai̇k Gürsoy, Vivek Kumar

Turkish Journal of Mathematics

The Jungck--Khan iterative scheme for a pair of nonself operators contains as a special case Jungck--Ishikawa and Jungck--Mann iterative schemes. In this paper, we establish improved results about convergence, stability, and data dependence for the Jungck--Khan iterative scheme.

On The Stability Of Cycles By Delayed Feedback Control, Dmitriy Dmitrishin, Paul Hagelstein, Anna Khamitova, Alexander M. Stokolos

Mathematical Sciences: Faculty Publications

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizingT-cycles of a differentiable functionf:R→Rof the form

x(k+1)=f(x(k))+u(k)

where

u(k)=(a₁−1)f(x(k))+a₂f(x(k−T))+...+a_Nf(x(k−(N−1)T)),

with a₁+...+a_N=1. Following an approach of Morgül, we construct a map F:R^T+1→R^T+1 whose fixed points correspond to T-cycles of f. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of F. We associate to each periodic orbit of f an …

Analysis Of Multiple Collision-Based Periodic Orbits In Dimension Higher Than One, Skyler C. Simmons

Theses and Dissertations

We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur

Turkish Journal of Mathematics

The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions.

Stability Of Compact Ricci Solitons Under Ricci Flow, Mina Vaghef, Asadollah Razavi

Turkish Journal of Mathematics

In this paper we establish stability results for Ricci solitons under the Ricci flow, i.e. small perturbations of the Ricci soliton result in small variations in the solution under Ricci flow.

Stability Analysis Of 4-Species Aβ Aggregation Model: A Novel Approach To Obtaining Physically Meaningful Rate Constants, G. Ghag, P. Ghosh, A. Mauro, V. Rangachari, Ashuwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

Protein misfolding and concomitant aggregation towards amyloid formation is the underlying biochemical commonality among a wide range of human pathologies. Amyloid formation involves the conversion of proteins from their native monomeric states (intrinsically disordered or globular) to well-organized, fibrillar aggregates in a nucleation-dependent manner. Understanding the mechanism of aggregation is important not only to gain better insight into amyloid pathology but also to simulate and predict molecular pathways. One of the main impediments in doing so is the stochastic nature of interactions that impedes thorough experimental characterization and the development of meaningful insights. In this study, we have utilized a …

A State-Dependent Delay Equation With Negative Feedback And ‘Mildly Unstable’ Rapidly Oscillating Periodic Solutions, Benjamin B. Kennedy

Math Faculty Publications

We consider state-dependent delay equations of the form x′(t)=f(x(t−d(x(t)))) where d is smooth and f is smooth, bounded, nonincreasing, and satisfies the negative feedback condition xf(x)x≠0. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution p. The initial segment p0 of p is the fixed point of a return map R that is differentiable in an appropriate setting.

We show that, although all the periodic solutions p we consider are unstable, the stability can be made arbitrarily mild in the sense that, given ϵ>0, we can choose f and d such …

A Nonlocal Parabolic Problem In An Annulus For The Heaviside Function In Ohmic Heating, Fei Liang, Hongjun Gao, Charles Bu

Turkish Journal of Mathematics

In this paper, we consider the nonlocal parabolic equation u_t=\Delta u+\frac{\lambda H(1-u)}{\big(\int_{A_{\rho, R}} H(1-u)dx\big)^2}, x\in A_{\rho, R} \subset R^2, t>0, with a homogeneous Dirichlet boundary condition, where \lambda is a positive parameter, H is the Heaviside function and A_{\rho, R} is an annulus. It is shown for the radial symmetric case that: there exist two critical values \lambda_* and \lambda^*, so that for 0

Resonant Solutions And Turning Points In An Elliptic Problem With Oscillatory Boundary Conditions, Alfonso Castro, Rosa Pardo

All HMC Faculty Publications and Research

We consider the elliptic equation -Δu + u = 0 with nonlinear boundary conditions ∂u/∂n = λu + g(λ,x,u), where the nonlinear term g is oscillatory and satisfies g(λ,x,s)/s→0 as |s|→0. We provide sufficient conditions on g for the existence of sequences of resonant solutions and turning points accumulating to zero.

Mean-Field Boolean Network Model Of A Signal Transduction Network, Naomi Kochi, Mihaela Teodora Matache

Mathematics Faculty Publications

In this paper we provide a mean-field Boolean network model for a signal transduction network of a generic fibroblast cell. The network consists of several main signaling pathways, including the receptor tyrosine kinase, the G-protein coupled receptor, and the Integrin signaling pathway. The network consists of 130 nodes, each representing a signaling molecule (mainly proteins). Nodes are governed by Boolean dynamics including canalizing functions as well as totalistic Boolean functions that depend only on the overall fraction of active nodes. We categorize the Boolean functions into several different classes. Using a mean-field approach we generate a mathematical formula for the …

Nested Canalyzing Depth And Network Stability, Lori Layne, Elena Dimitrova, Matthew Macauley

Publications

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input …

A Generalized Cholera Model And Epidemic-Endemic Analysis, Jin Wang, Shu Liao

Mathematics & Statistics Faculty Publications

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R₀ = 1 for the combined human-environment epidemiological …

Bifurcation And Invariant Manifolds Of The Logistic Competition Model, M. Guzowska, Rafael Luís, Saber Elaydi

Mathematics Faculty Research

In this paper we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important center manifolds, and study their bifurcation. Saddle-node and period doubling bifurcation route to chaos is exhibited via numerical simulations.

The Principle Of Linearized Stability For Size-Structured Population Models, M. El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

The principle of linearized stability for size-structured population dynamics models is proved giving validity to previous stability results reported in, for example, El-Doma (2008-1). In particular, we show that if all the roots of the characteristic equation lie to the left of the imaginary axis then the steady state is locally exponentially stable, and on the other hand, if there is at least one root that lies to the right of the imaginary axis then the steady state is unstable. We also point out cases when there is resonance

Using Feed-Forward Networks To Infer The Activity Of Feedback Neuronal Networks, Xinxian Huang

Dissertations

The nervous system is one of the most important organ systems in a multicellular body. Animals, including human beings perceive, learn, think and deliver motion instructions through their nervous system. The basic structural units of the nervous system are individual neurons which constitute different neuronal networks with distinct functions. In each network, constituent neurons are coupled with different connection patterns, for example, some neurons send feed-forward information to the coupling neurons while others are mutually coupled. Because it is often difficult to analyze large interconnected feedback neuronal networks, it is important to derive techniques to reduce the complexity of the …

A Stability Study Of A New Explicit Numerical Scheme For A System Of Differential Equations With A Large Skew-Symmetric Component, Katharine Gurski, Stephen O'Sullivan

Articles

Abstract. Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time-step, but require more computations. When the differential equations are dominated by a skew-symmetric component, the problem is not stiffness in the sense that the size of the eigenvalues are unequal, rather the that the real eigenvalues are dominated by imaginary eigenvalues. We present and compare analytical results for stable time step limits for several explicit methods including the super-time-stepping method of Alexiades, Amiez, and Gremaud …

Towards A Theory Of Periodic Difference Equations And Its Application To Population Dynamics, Saber Elaydi, Rafael Luís, Henrique Oliveira

Mathematics Faculty Research

This survey contains the most updated results on the dynamics of periodic difference equations or discrete dynamical systems this time. Our focus will be on stability theory, bifurcation theory, and on the effect of periodic forcing on the welfare of the population (attenuance versus resonance). Moreover, the survey alludes to two more types of dynamical systems, namely, almost periodic difference equations and stochastic di®erence equations.

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

Analytic & Numerical Study Of A Vortex Motion Equation, Daniel Bueller

Electronic Theses and Dissertations

A nonlinear second order differential equation related to vortex motion is derived. This equation is analyzed using various numerical and analytical techniques including finding approximate solutions using a perturbative approach.

Non-Autonomous Periodic Systems With Allee Effects, Rafael Luís, Saber Elaydi, Henrique Oliveira

Mathematics Faculty Research

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper the properties and stability of the three fixed points are studied in the setting of nonautonomous periodic dynamical systems or difference equations. Finally we investigate the bifurcation of …

Population Models With Allee Effect: A New Model, Saber Elaydi, Robert J. Sacker

Mathematics Faculty Research

In this paper we develop several mathematical models of Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model that is our focus of study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.

Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski

All HMC Faculty Publications and Research

A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The …

Linear Stochastic Systems: A White Noise Approach, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study ℓ1-ℓ2 stability in the discrete time case, and L2-L∞ stability in the continuous time case.

Forced Oscillations Of The Korteweg-De Vries Equation On A Bounded Domain And Their Stability, Muhammad Usman, Bingyu Zhang

Mathematics Faculty Publications

It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem

uu_xu_xxx 0 < x < 1, t > 0, (*)

It is shown …

An Economical Model With Allee Effect, Rafael Luís, Saber Elaydi, Henrique Oliveira

Mathematics Faculty Research

The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hypothesis of Marx …

Stability And Asymptoticity Of Volterra Difference Equations: A Progress Report, Saber Elaydi

Mathematics Faculty Research

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Ζ-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried …

Criteria Pollutant Concentration, Victor Neese

Undergraduate Journal of Mathematical Modeling: One + Two

With the assistance of Assistant Professor Jeff Cunningham (Civil & Environmental Engineering Department, USF) a study was carried out pertaining to the amount of CO and NOx pollutants emitted from a highway having two lanes in each direction. The receptor of the pollutants is a hypothetical housing development located near the highway. The concentrations were found by utilizing information regarding meteorological conditions, estimated emissions rates, and distances from the highway less than or equaling 2 km.

The Stabilizing Effect Of Noise On The Dynamics Of A Boolean Network, Christopher S. Goodrich, Mihaela Teodora Matache

Mathematics Faculty Publications

In this paper, we explore both numerically and analytically the robustness of a synchronous Boolean network governed by rule 126 of cellular automata. In particular, we explore whether or not the introduction of noise into the system has any discernable effect on the evolution of the system. This noise is introduced by changing the states of a given number of nodes in the system according to certain rules. New mathematical models are developed for this purpose. We use MATLAB to run the numerical simulations including iterations of the real system and the model, computation of Lyapunov exponents, and generation of …