Session II.1 - Computational Dynamics
Talks
Thursday, June 15, 14:00 ~ 14:30
Validated integration of semilinear parabolic PDEs
Maxime Breden
Ecole polytechnique, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this talk, we present a computer-assisted proof methodology to perform rigorous time integration for semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variations of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition in time, we then finish the proof with a Newton-Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift-Hohenberg equation, the Ohta-Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.
Joint work with Jan Bouwe van den Berg (VU Amsterdam) and Ray Sheombarsing (VU Amsterdam).
Thursday, June 15, 14:30 ~ 15:00
Symbolic dynamics and abundance of connecting orbits between periodic orbits in the Kuramoto-Sivashinsky PDE
Daniel Wilczak
Jagiellonian University, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
For some parameter value for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we prove the existence of infinite number of homo- and heteroclinic orbits to two periodic orbits. The proof is computer assisted. We present a new algorithm for the rigorous integration of the variational equation (i.e. producing $C^1$ estimates) for a class of dissipative PDEs on the torus.
Joint work with Piotr Zgliczyński (Jagiellonian University, Poland).
Thursday, June 15, 15:00 ~ 16:00
Cyclic Symmetry Induced Pitchfork Bifurcations in the Diblock Copolymer Model
Evelyn Sander
George Mason University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcation points. Many of these bifurcations are of pitchfork type, and in previous work we showed that pitchfork bifurcations are induced by a simple even or odd symmetry-breaking and can be validated using computer assisted proof. However, this is not the complete picture: many other pitchfork bifurcations do not exhibit even or odd symmetry breaking. In the current work, we show that in these more involved cases, a cyclic group action is responsible for their existence, based on cyclic groups of even order. We present theoretical results establishing such bifurcation points and show that they can be characterized as nondegenerate solutions of a suitable extended nonlinear system, and show how these results can be validated using computer assisted proofs.
Joint work with Thomas Wanner (George Mason University) and Peter Rizzi (George Mason University).
Thursday, June 15, 16:30 ~ 17:00
Computation of invariant tori in close-to-degenerate Hamiltonian systems
Alex Haro
Universitat de Barcelona & CRM, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Motivated by Celestial Mechanics problems, in this talk we present methods for computing invariant tori in close-to-degenerate Hamiltonian systems. In particular, we present quasi-Newton methods for the invariance equation of the torus parameterization, leading to proofs of new KAM results, numerical algorithms, and computer-assisted proofs.
Joint work with Jordi-Lluís Figueras (Uppsala University, Sweden).
Thursday, June 15, 17:00 ~ 17:30
Towards computer assisted proofs based on combinatorial multivector fields
Marian Mrozek
Jagiellonian University, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
The family of strongly connected components of a directed graph constructed from a transversal cellular decomposition of a flow provides an algorithmic tool in the automated rigorous analysis of the gradient structure of a dynamical system. However, it is less helpful in the study of recurrent dynamics. A combinatorial multivector field may be viewed as a directed graph whose set of vertices is a topological space. In the talk I will present some recent results based on combinatorial multivector fields which indicate their potential in automated rigorous analysis of recurrent dynamics.
Friday, June 16, 14:00 ~ 14:30
Forcing results for travelling waves in a cylinder through CAPs for equilibria
Jan Bouwe van den Berg
VU Amsterdam, Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
Travelling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. A topological invariant, based on a Floer homology construction, gives insight into the structure of the solutions of the reaction-diffusion equations. It encodes relations between connecting orbits on the one hand and equilibria, enhanced with (relative) index information, on the other. This leads to forcing theorems for travelling wave solutions. These theorems can be made effective by finding equilibria and their indices through computer-assisted proofs.
Friday, June 16, 14:30 ~ 15:00
Validation of codimension 2 bifurcations in DDEs
Elena Queirolo
TUM , Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Arc-length continuation is well suited to follow 1 dimensional curves and has been successfully coupled with the radii polynomial approach to validate solutions to parameter dependent problems. In this talk, we discuss algorithms to follow 2-dimensional manifolds originated by problems with multiple parameters. Then, we couple this numerical method with a broader formulation of the radii polynomials to allow for the validation of this computed manifold. In particular, we look at ODEs and DDEs that undergo codimension 2 bifurcations of the saddle-hope and Bautin type. To study these systems we apply a global blow up and track both the periodic orbits and the fixed points in a neighbourhood of the codim2 bifurcation.
Joint work with Kevin Church (McGill).
Friday, June 16, 15:00 ~ 16:00
Computer algebra and chaotic diffusion of planetary motions in the solar system.
Jacques Laskar
Observatoire de Paris, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
The evidence of the chaotic movement of the planets in the solar system was obtained thanks to a numerical integration of the averaged equations of motion (Laskar, 1989). This system of equations containing more than 150,000 terms had been obtained by very dedicated methods of computer algebra, whose adaptation to other problems was not easy. Since 1988 began the construction of a general computer algebra system, TRIP, specially adapted to celestial mechanics calculations and perturbation methods. We have recently used this system to get a better understanding of the origin of chaos in the solar system, and to study the chaotic diffusion of the motion of the planets over times well in excess of the age of the universe.
Friday, June 16, 16:30 ~ 17:00
Convective periodic axisymmetric flows in rotating fluid spheres: from their onset to their stability.
Juan Sánchez Umbría
Universitat Politècnica de Catalunya, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
The onset of convection in rotating fluid spheres and shells usually gives rise to rotating waves, which can travel in the prograde or retrograde direction relative to the frame of reference rotating with the bulk of the fluid. It was discovered recently that axisymmetric periodic regimes can also be preferred at low Prandtl, Pr, and Ekman, Ek, numbers. These flows are known as torsional.
In order to determine the region in the parameter space where the torsional flows are the first bifurcated solutions, the curves of double Hopf points corresponding to simultaneous transitions to azimuthal wave numbers $(m_1,m_2)$=(0,1), (1,1), (0,2), etc. were computed. These curves form the skeleton of the bifurcation diagram, separating the regions of different preferred azimuthal wave numbers. Their intersections are triple Hopf points, several of which were found. It turned out that the region of interest was limited by the curves $(m_1,m_2)$=(0,1) and (0,2).
The periodic torsional solutions emerging form the conduction state were computed for several pairs (Pr,Ek) inside the above mentioned region, using Newton-Krylov continuation methods, and their stability to azimuthal dependence was studied via their Floquet multipliers.
Joint work with Marta Net (Universitat Politècnica de Catalunya).
Friday, June 16, 17:00 ~ 17:30
Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: computer-assisted proofs of existence
Jean-Philippe Lessard
McGill University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we present a general method to rigorously prove existence of strong solutions to a large class of semi-linear PDEs in Sobolev/Hilbert spaces via computer-assisted proofs. Considering a large enough hypercube, we use Fourier series to compute a numerical approximation of the solution, which is then refined via a finite-dimensional trace theorem to obtain a smooth function with support on the hypercube. Finally, a Newton-Kantorovich theorem is applied to demonstrate that a true solution exists nearby this refined solution. As an application, we prove the existence of stationary non-radial localized patterns in the planar Swift-Hohenberg PDE.
Saturday, June 17, 14:00 ~ 14:30
Algorithmic approach to the global dynamics of multi-parameter systems of ODEs
Konstantin Mischaikow
Rutgers University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss an algorithmic approach for identifying the global dynamics of multi-parameter systems of ODEs. We are far from achieving this goal. However our initial effort requires a variety of techniques and arguments, and thus I will limit myself to a few major points: (1) I will discuss what we mean by solve, argue that a non-traditional notion of solution is necessary, and suggest one based on order theory and algebraic topology. (2) I will discuss the philosophy of the approach we are taking. (3) I will introduce a specific family of differential equations (ramp systems) for which we can produce a combinatorial representation of the dynamics, a well defined finite decomposition of parameter space, and show results from some of the computations that can be done currently.
This is ongoing work with W. Duncan, D. Gameiro, M. Gameiro, T. Gedeon, H. Kokubu, H. Oka, B. Rivas, and E. Vieira.
Saturday, June 17, 14:30 ~ 15:00
Computing the Global Dynamics of Parameterized Systems of ODEs
Marcio Gameiro
Rutgers University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics is extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to systems of ODEs arising from gene regulatory networks.
Saturday, June 17, 15:00 ~ 15:30
Network Dynamics Modeling and Analysis (NDMA): A flexible Python library for analysis of network dynamical systems
Shane Kepley
Vrije Universiteit, Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a Python based framework for analyzing dynamical phenomena in network dynamical systems. This framework enables one to study a wide variety of models given only a network topology and activation functions as input. Examples of its capabilities include efficiently bounding, finding equilibria and isolating equilibria, finding saddle-node bifurcations, and Lagrangian optimization all in high dimensional parameter models. We demonstrate the library's capabilities with some examples from Systems Biology.
Joint work with Elena Quierolo (Technische Universität München).
Saturday, June 17, 15:30 ~ 16:00
Persistence and computation of some invariant objects in functional perturbations of an ODE
Joan Gimeno
Universitat de Barcelona, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
With very minor assumptions, I show that periodic orbits and hyperbolic sets in an ODE persist under singular perturbations of including a delay term. These perturbations change the phase space from finite to infinite dimensions. The results apply to electrodynamics and give new approaches to handle state-dependent, small, nested, and distributed delays.
I will discuss and explain motivations, the new methods, sketches of the proofs, computer assisted proofs, and possible applications. I will end the talk giving some ideas of work in progress and possible future works.
Joint work with Jiaqi Yang (Clarkson University, USA) and Rafael de la Llave (Georgia Institute of Technology, USA).
Saturday, June 17, 16:30 ~ 17:00
Central configurations - some rigorous computer assisted results
Piotr Zgliczynski
Jagiellonian University, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will give an overview of our of recent computer assisted proofs for the rigorous count of central configurations.
Our approach is based on: - the use of interval arithmetics methods, for example the Newton-Krawczyk operator - a priori bounds for central configurations
This allows to obtain an rigorous listing of all central configurations when masses are away from zero and there are no bifurcation nearby in the mass space, we have done for equal masses in the planar case for $n=5,6,7$ and in the spatial case for $n=5,6$.
To extend this approach to all masses the following issues has to be solved: - understanding of restricted N+k problems (N-big masses and k "massless" bodies) and their continuation to full problem - rigorous treatment of bifurcations
References:
1) M.~Moczurad, P.~Zgliczy\'nski, Central configurations in planar $n$-body problem for $n=5,6,7$ with equal masses, arXiv:1812.07279, Celestial Mechanics and Dynamical Astronomy, (2019) 131: 46,
2) M.~Moczurad, P.~Zgliczy\'nski, Central configurations in spatial $n$-body problem for $n=5,6$ with equal masses }, Celestial Mechanics and Dynamical Astronomy, (2020) 132:56
3) M.~Moczurad, P.~Zgliczy\'nski, Central configurations on the plane with $N$ heavy and $k$ light bodies, Communications in Nonlinear Science and Numerical Simulation, 114 (2022), 106533
Joint work with Moczurad Malgorzata (Jagiellonian University, Poland).
Saturday, June 17, 17:00 ~ 17:30
Asymptotic scaling and universality for skew products with factors in SL(2,R)
Hans Koch
The Universiy of Texas at Austin, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the asymptotic properties of matrix products that appear in the study of the Hofstadter model (of an electron moving on a lattice) and the associated almost Mathieu operators. In a restricted setup that is characterized by a symmetry, we show that critical behavior occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. Other periodic orbits govern supercritical behavior.
Posters
Modeling oligomers-prion interaction using delay differential equations
yazid bensid
ESSA-Tlemcen, LBM, Sidi belabbes university, Algérie - This email address is being protected from spambots. You need JavaScript enabled to view it.
Alzheimer's disease (AD) is a progressive neurodegenerative disorder characterized by cognitive decline and memory impairment. The exact mechanisms underlying AD are not yet fully understood, but it is known that Amyloid beta oligomers, which are small aggregates of A$\beta$ protein, play crucial roles in the pathogenesis of this disease . Recent studies suggest that A$\beta$ oligomers tend to bind to healthy prions and misfold the latters into a pathogenic form called $PrP^{ol}$. In this work we present a model that tries to describe the interaction between A$\beta$ oligomers and prions. we investigate equilibria and study their local stability.
$$\dot U(t)=S_u-d_u U(t)-2\delta_1 P(t)U^2(t)+2\delta_{2} P(t-\tau)U^2(t-\tau)$$ $$\dot P(t)=S_p-d_p P(t)-\delta_1 P(t)U^2(t)$$ $$\dot P^{ol}(t)=-d_{pc} P_c(t)+\delta_{2} P(t-\tau)U^2(t-\tau)$$
Where $U,P$ and $ P^{ol}$ denote the concentrations in A$\beta$ oligomers, prions and pathogenic prions respectively.
Joint work with Mohamed Helal ( LBM, Sidi belabbes university) and Abdelkader Lakmeche ( LBM, Sidi belabbes university).
Attracting period 3 implies all natural periods for multidimensional maps
Anna Gierzkiewicz
Jagiellonian University in Krakow, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present the method from [ A. G., P. Zgliczynski, J Differ Equ, 314 (2022),733--751] for finding a wide variety of periodic orbits for multidimensional maps with an attracting $n$-periodic orbit. The set of periods is induced by the Sharkovskii ordering '$\triangleleft$' of natural numbers: \[ 3\triangleleft 5 \triangleleft 7 \triangleleft \cdots \triangleleft 2\cdot 3 \triangleleft 2 \cdot 5 \triangleleft \cdots \triangleleft 2^2\cdot 3 \triangleleft 2^2 \cdot 5 \triangleleft \dots \triangleleft 2^k \triangleleft 2^{k-1} \triangleleft \cdots \triangleleft 2^2 \triangleleft 2 \triangleleft 1\text{.} \]
As an example, we prove the existence of $n$-periodic orbits for all $n\in\mathbb{N}$ in the Roessler system with a $3$-periodic orbit, the existence of $n$-periodic orbits for all $n\in\mathbb{N}\setminus\{3\}$ in a similar system with a $5$-periodic orbit, etc. We also expect that this method works for DDEs. The proofs are computer-assisted with the use of CAPD library for C++.
Joint work with Piotr Zgliczynski (Jagiellonian University, Poland).
Numerical method for solving special Cauchy problem for the second order integro-differential equation
Sayakhat Karakenova
Kh. Dosmukhamedov Atyrau University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the present communication we conserved the special Cauchy problem for the second order integro-differential equation $$ \frac{d^2x}{dt^2}=A_1(t){\frac{dx}{dt}} +A_2(t) x(t)+\varphi_1(t) {\int_{0}^{T}} \psi_1(\tau) f_1(\tau,\dot{x}(\tau))d\tau +\varphi_2(t) {\int_{0}^{T}} \psi_2(\tau) f_2(\tau,x(\tau))d\tau+g(t) \;(1) $$
where the $A_1(t), A_2(t), \varphi_1(t), \varphi_2(t),\psi_1(\tau), \psi_2(\tau)$ are continuous on $f:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n$, is continuous.
A solution to equation (1) is continuosly differentable on [0,T] function $x(t)\in C([0,T],R^n)$, which satisfies equation for any $t \in [0,T]$.
Equation (1) adduce to special Cauchy problem by the Dzhumabaev parametrization method. An iterative method is proposed to solve a special Cauchy problem. The iterative method is implemented numerically.
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP15473218).
Numerical simulation of no-slope-selection epitaxial thin film growth
Hyun Geun Lee
Kwangwoon University, Republic of Korea - This email address is being protected from spambots. You need JavaScript enabled to view it.
To simulate the no-slope-selection epitaxial thin film growth numerically, we construct a numerical method for the growth equation by combining the linear convex splitting with the second-order strong-stability-preserving implicit-explicit Runge-Kutta method. As a result, the method is linear, accurate, and unconditionally energy stable. Using the method, we perform the long time simulation for the coarsening process, where the $-\ln(t)$ energy decay and $t^{1/2}$ roughness growth rates can be observed clearly.
Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times
Meirambek Mukash
Aktobe Regional University named after K. Zhubanov, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method were developed and applied for investigating of various problems. As well-known, impulsive systems of differential equations supply as mathematical models of objects that, in the course of their evolution, they are subjected to the action of short-term forces. Much research has been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions of impulsive systems also are considered.
In this communication, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and on infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.
In this paper, we consider a exact system of differential equations with impulsive effects at non-fixed times and a small parameter of the following form $$\dot{x}(t)=\varepsilon X(t,x), \qquad t\neq t_{i}(x), $$ $$\Delta{x}\Bigl{|}_{t=t_{i}(x)} =\varepsilon {I}_{i}(x)$$ $$x(0)=x_{0}$$
where $\varepsilon \gt 0$ is a small parameter, $t_{i}(x) \lt t_ {i +1}(x)$ $(i = 1,2,…)$ moments of impulsive effects, functions $X$ and $I_{i}$ $d$ are $n$ - dimensional vector of functions.
We put $U_{a} = \{x\in R^d: | x |\leq a\}$. Suppose the following conditions are met:
1. The functions $X(t,x)$ and $I_{i}(x)$ are continuous in the set $Q = \{t\geq 0,x\in U_{a}\}$, bounded by a constant $M \gt 0$, and in $x$ satisfy the Lipschitz condition with a constant $L \gt 0$;
2. Uniformly in $t,x$ for $t\geq 0,x\in U_{a}$, there exist finite limits $$X_{0}(x)=\lim_{T \to \infty}\frac{1}{T}\int ^{t+T}_t X(s,x)ds,$$ $$I_{0}(x)=\lim_{T \to \infty}\frac{1}{T}\sum \limits_{t \lt t_{i}(x) \lt T}I_{i}(x),$$
3. Solution $y=y(t), y(0)= x(0)$ of the averaged system $$\dot{y}=\varepsilon[X_{0}(y) + I_{0}(y)] $$ is defined for $t\geq0$ and lies in $U_{a}$ together with some neighborhood $\rho$ and is uniformly asymptotically stable;
4. The moments of the impulsive effect ${t_{i}(x)}$ are continuously and their functions satisfy in $U_{a}$ uniformly in $i\in N$, and the surfaces $t = t_{i}(x)$ satisfy the separation condition, that is $$\min\limits_{x\in U_{a}} t_{i+1}(x) \lt \min\limits_{x\in U_{a}} t_{i}(x) (i = 1,2,…)$$
Suppose that there is a constant $C \gt 0$ such that for all $t \gt 0$ and $x\in U_{a}$ $$i(t, x)\leq Ct $$ where $i(t, x)$ is the number of pulses on $(0,t)$.
It is also assumed that the solutions of exact system intersect each surface $t = t_{i}(x)$ at most once, that is, there is no beating. The conditions for the absence of beating are well studied.
Theorem 1. Let Conditions 1-4 be satisfied. Then, for an arbitrary $\eta \gt 0$, one can specify $\varepsilon_{0}$ such that $\varepsilon \lt \varepsilon_{0}$ for $t\geq0$, the inequality
$$ |x(t)-y(t)| \lt \eta $$
where $x(t) (x(0) = y(0) = x_{0})$ is a solution to the exact system .
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP 15473190)
Computational method for solving a boundary value problem for an impulsive integro-differential equation
Sandugash Mynbayeva
K.Zhubanov Aktobe Regional University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this study, the boundary value problem for the Fredholm integro-differential equation subject to impulse effects at fixed time points is considered: $$ \frac{dy}{dt}=A(t)y +\sum\limits_{l=1}^{k}\int_{0}^{T}\varphi_l(t) \psi_l(s)y(s)ds+f(t), \, t\in (0, T) \setminus \{\theta_1, \theta_2, \ldots, \theta_m\}, \, y\in R^n, \eqno (1) $$ $$ (0=\theta_0 \lt \theta_1 \lt \ldots \lt \theta_m \lt \theta_{m+1}=T),$$ $$ By(0)+Cy(T)=d, \quad d\in R^{n}, \eqno (2) $$ $$ \Delta y(\theta_j)=\sum_{i=0}^{j-1}d_{ij}y(\theta_i+0), \quad j=\overline{1,m}, \eqno (3)$$ where $\Delta y(\theta_j)=y(\theta_j+0)-y(\theta_j-0),$ the square matrices $A(t)$, $\varphi_l(t),$ and $\psi_l(s)$ of order $n$ are continuous on $[0,T]$, $f(t)$ is piecewise continuous on $[0,T]$, with the possible exception of the points $t = \theta_j, j = \overline{1,m},$ the square matrices $B,$ $C,$ and $d_{ij}$ of order $n$ and the vector $d$ are constant.
A constructive method for solving the problem based on the Dzhumabaev para- metrization method is proposed, and an algorithm for finding a numerical solution to the problem is proposed. The algorithms consist of constructing and solving the systems of linear algebraic equations with respect to the arbitrary parameters. Since the Cauchy problems are solved independently from each other, their solutions can be found by parallel computing. In our calculations, we used the fourth-order Runge-Kutta method, the Adams and Bulirsch-Stoer methods for solving auxiliary Cauchy problems. We also used used Simpson’s rule for estimating definite integrals.
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP13268824).
Air-Mucus Interaction in a Pulmonary Airway: a perspective of a process engineer
Bharat Soni
University Côte d'Azur, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
A hybrid approach based on computational fluid dynamics (CFD) and lumped parameter-based modeling (LPM) is used to understand the comprehensive role of air-mucus interaction flow in the pulmonary airway. The mucus is modeled as Bingham fluid which can either acts as (1) partially fluid and partially liquid; (2) purely liquid and (3) purely solid. An equivalent impedance is derived that depends on the airway geometry, yield stress, and fluid parameters. It is observed that gravity plays a significant role in the transportation of mucus which is essential in airway clearance techniques, especially in case of respiratory diseases like asthma, COPD, cystic fibrosis, etc. where mucus is over-secreted. The derived impedance provides clear insights into the mucus over secretion.
Joint work with Benjamin Mauroy (University Côte d'Azur, France).
Stability Analysis of Fractional-Order Neural Network Model with Proportional Delay
Swati Tyagi Swati Tyagi
Amity University Punjab, Mohali, India - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, a fractional-order neural network with proportional time delay is examined for the existence of a unique solution. Sufficient conditions for the existence of a solution for the given model are derived with respect to various initial conditions using analysis of fixed point theory and fractional calculus. The analytical findings are explained through numerical examples. These models are suitable when a process takes place through strongly anomalous media.
Rigorous numerics, Poincaré maps, and Covering Relations in infinite-dimensional spaces for computer assisted proofs in Delay Differential Equations.
Robert Szczelina
Jagiellonian University, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will show some general principles of a recently developed high-order Lohner-type algorithm for a rigorous integration of systems of Delay Differential Equations (DDEs) [1], together with some topological tools in the infinite dimensional phase-space of DDEs that are suitable for computer assisted proofs.
We will use these tools to prove, with the computer assistance, various kinds of dynamical behaviour, for example, existence of several (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of classical values of parameters where chaos is numerically observed), persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system), and the rigorous computation of trajectories in piecewise defined DDEs.
The method is quite general and does not impose severe restrictions on the kind of solutions it can track, i.e. the integration time does not need to be a multiple of the basic time lag nor the solutions need not to be of a specific class, e.g. periodic.
[1] R. Szczelina and P. Zgliczyński. High-order Lohner-type algorithm for rigorous computation of Poincaré maps in systems of Delay Differential Equations with several delays, {\em Accepted for publication in Found. Comput. Math., preprint: https://arxiv.org/abs/2206.13873}, (2023).
Joint work with Piotr Zgliczyński (Jagiellonian University).
Numerical solution of a semi-periodic initial problem of the fourth order hyperbolic type using a generalized operator and functional parametrization
Zhanibek Tokmurzin
K.Zhubanov Aktobe Regional University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
The domain $\Omega=[0,T]\times[0,\omega]$ we consider the following semi-periodic initial boundary value problem for a fourth order system of partial differential equations: $$ \frac{\partial^4 u}{\partial t^3 \partial x} =A_1 (t,x) \frac{\partial^3 u}{\partial t^2 \partial x}+A_2 (t,x) \frac{\partial^3 u}{\partial t^3 }+A_3 (t,x) \frac{\partial^2 u}{\partial t^2}+A_4 (t,x) \frac{\partial^2 u}{\partial t \partial x}+$$ $$ +A_5 (t,x) \frac{\partial u}{\partial t}+A_6 (t,x)\frac{\partial u}{\partial x} +A_7 (t,x)u+f(t,x), $$ $$ u(0,x)=\varphi_1 (x), \quad x\in[0,\omega], $$ $$ \frac{\partial u(t,x)}{\partial t}\Big|_{t=0} =\varphi_2 (x), \quad x\in[0,\omega], )$$ $$ \frac{\partial^2 u(t,x)}{\partial^2 t}\Big|_{t=0} =\frac{\partial^2 u(t,x)}{\partial^2 t}\Big|_{t=T}, \quad x\in[0,\omega], $$ $$ u(t,0)=\psi (x), \quad t\in[0,T], $$ where $u(t,x)=col(u_1 (t,x),u_2 (t,x),…,u_n (t,x))$ is the unknown function; the $n\times n$-matrices $A_i (t,x), (i=\overline{1,7}),$ and $n$- vector function $f(t,x)$ are continuous on $ \Omega$; $n$ vector-function $\psi(t)$ are continuously three times differentiable on $[0,T]$; the $n$ vector-functions $\varphi_1 (x)$ and $\varphi_2 (x)$ are continuously differentiable on $[0,\omega]$.
Reducing the order of the problem by two times, we reduce it to the Cauchy problem for a system of n first-order ordinary differential equations. It is shown that the Cauchy problem has a solution using the method of generalized operations or the method of functional parameterization. Using numerical methods for one variable of the Cauchy problem, solutions of the Cauchy problem are obtained.
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14971198).
Solving the stochastic Helmholtz problem by the method of moment functions
Gulmira Vassilina
Institute of Mathematics and Mathematical Modeling, Almaty University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we study the solvability of the stochastic Helmholtz problem in the sense of the equivalence of equations in the mean and in the quadratic mean. We consider the linear statement of the problem. This is explained by the efficiency of the method of moment functions as applied to linear equations, whereas the efficiency of this method markedly decreases when applied to nonlinear equations. The essence of the method of moment functions is that it reduces the study of a stochastic equation to the study of a system of ordinary differential equations with respect to the moments under consideration. The given the system of second-order stochastic Ito equations linear in drift \[\ddot{x}_{i} =a_{ik} (t)\dot{x}_{k} +b_{ik} (t)x_{k} +\sigma _{ij} (t,x,\dot{x})\dot{\xi }^{j} ,\quad (i=\overline{1,n};j=\overline{1,m})., \eqno{(1)}\] reduce it to equivalent equations of the Lagrangian structure. $\xi =(\xi _{1} (t,\omega ),\ldots ,$ $ \xi _{r} (t,\omega ))^{T} $ is a system of random processes with independent increments. To solve the problem, we apply the operation of mathematical expectation $M(\cdot )$ to Eq. (1). By introducing the denotation $m_{\nu } (t)=Mx_{\nu } (t)$, we obtain the equation \[\ddot{m}_{i} =a_{ik} (t)\dot{m}_{k} +b_{ik} (t)m_{k} ,\quad (i=\overline{1,n};\; k=\overline{1,n}). \eqno{(2)}\] We now formulate the indirect Helmholtz problem in the space $(m,\dot{m})$ as follows: given Eq. (2), determine the conditions for the functions $h_{i}^{\nu } $ and the Lagrange function $L=L(m,\dot{m},t)$, under which the following relations hold \[ h_{i}^{\nu } (\ddot{m}_{i} -a_{ik} (t)\dot{m}_{k} -b_{ik} (t)m_{k} )=\frac{d\; }{dt} (\frac{\partial L}{\partial \dot{m}_{i} } )-\frac{\partial L}{\partial m_{i} } . \] We construct the equations of the Lagrangian structure by the given second-order linear stochastic Ito equations in the spaces of moment functions of both the first and second order. Necessary and sufficient conditions for the direct and indirect representation of the Lagrangian are obtained. This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP09258966).
Joint work with Marat Tleubergenov (Institute of Mathematics and Mathematical Modeling, Kazakhstan).
Parameter estimation and uncertainty quantification for a mathematical model of breast cancer
Hsiu-Chuan Wei
Feng Chia University, Taiwan - This email address is being protected from spambots. You need JavaScript enabled to view it.
Breast cancer is the most commonly diagnosed cancer in women and the second leading cause of cancer death for women worldwide. Mathematical modeling is a powerful tool that helps us understand complex phenomena of biological systems. A mathematical model was previously constructed, using an ODE (ordinary differential equation) system, to study the tumor growth in breast cancer [1]. The parameter values were estimated using experimental data published in scientific literature. However, parameter estimation of dynamical systems is often associated with parameter uncertainty, and the analysis of parameter uncertainty is important as it determines how reliable the model outputs are. In this study, the mathematical model proposed in [1] is revisited. Uncertainty quantification is carried out using the Markov chain Monte Carlo method, and the effect of parameter uncertainty on the dynamics of the model is addressed.
[1] Hsiu-Chuan Wei, Mathematical modeling of tumor growth: the MCF-7 breast cancer cell line, Mathematical Biosciences and Engineering, 16(6), 2019, 6515-6535, DOI: 10.3934/mbe.2019325.