Special Session 138: Recent advances in Fractal Geometry, Dynamical Systems, and Positive Operators

Investigating the solutions of singular differential equations via Lucas wavelet artificial neural networks

Shivani Aeri Central University of Himachal Pradesh India Co-Author(s):    Shivani Aeri, and Rakesh Kumar

Abstract: Singular differential equations are essential for modeling complex phenomena in fields such as fluid dynamics, astronomy, biology, and physics. The Emden-Fowler model, Lane-Emden model, and Thomas-Fermi model are prominent singular models with extensive applications in thermodynamics, astrophysics, and atomic physics. These equations are complex due to the presence of singularities, which makes them difficult to solve analytically. As exact solutions are not always achievable, numerical approaches are crucial for estimating solutions to these equations. Developing robust numerical approaches is vital for addressing the issues associated with singularities. This study investigates the development of the Lucas wavelet artificial neural network model and its application in solving singular differential equations. This approach integrates the efficacy of wavelet theory with the flexibility of artificial neural networks. The wavelet method`s capacity to encapsulate both local and global attributes of the solution is beneficial. Moreover, the neural network approach offers some advantages over alternative numerical methods for managing intricate non-linear models with enhanced efficiency. Comparative analyses demonstrate that this methodology yields accurate and consistent results, making it a powerful tool for scientific and technical problems.

Dimension of set-valued functions and their distance sets

Ekta Agrawal Indian Institute of Information Technology Allahabad India Co-Author(s):    Dr. Saurabh Verma

Abstract: Barnsley introduced the theory of non-smooth interpolation for finite data in his seminal work ``Fractal Functions and Interpolation, Constr. Approx 2 (1986) 303-329 by exploiting an iterated function system concept. After that, numerous fractal functions are constructed corresponding to real/vector-valued functions. Recently, fractal interpolation functions corresponding to a set-valued function on a compact interval of the real line have been constructed. Set-valued functions played a significant role in applied areas such as mathematical modeling, game theory, control theory, and many more. Dimension estimation of any set or the graph of a function remains a vibrant area of research in the literature. In this talk, we first construct the set-valued fractal function corresponding to any continuous set-valued function defined on the compact subset of the real line using the metric sum of sets. Subsequently, some results are obtained for the bounds estimation of fractal dimensions, such as the Hausdorff and box dimension for the graphs of constructed fractal functions. Further, some bounds on the dimension of the distance sets of graphs of these functions are also discussed. In the end, we shed some light on the celebrated Falconer`s distance-set conjecture regarding the graphs of set-valued functions.

On construction of fractal functions and fractal measures

Subhash Chandra HRI India Co-Author(s):

Abstract: In this talk, we discuss a novel method to construct fractal functions with graphical illustrations. Fractal measures play an important role in the theory of fractal dimensions. We also show the existence of fractal measures supported by the attractor of the iterated function system satisfying the strong separation condition.

Various Notions of Shadowing in Triangular Map

DEEPANSHU DHAWAN INDIAN INSTITUTE OF TECHNOLOGY JODHPUR India Co-Author(s):    Dr. Puneet Sharma

Abstract: In this talk, we will discuss various forms of shadowing for a general triangular system. In particular, we will relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. Furthermore, we will explore chain transitivity and chain mixing for a general triangular system. In addition, we investigate the connection between the dynamics of the non-autonomous component systems in a triangular system and also provide examples to demonstrate the necessity of the conditions imposed.

Fractals in sea ice dynamics

Ken Golden University of Utah USA Co-Author(s):    Kenneth M. Golden

Abstract: Polar sea ice is a critical component of Earth`s climate system. As a material it displays composite structure over a vast range of length scales. In fact, fractals appear naturally throughout the sea ice system, from brine inclusions inside the ice and labyrinthine melt ponds on its surface, to the ice pack itself on the scale of the Arctic Ocean. We explore how the fractal dimension of these structures depends on the parameters characterizing their dynamical evolution. These investigations lead us into percolation theory, statistical physics, and topological data analysis. In related work, we consider a dynamical systems model of algae living in the brine inclusions, with model parameters treated as random variables. Uncertainty quantification methods are used to study bloom dynamics.

$\alpha$-fractal operator on a subset of the first Heisenberg group

Gurubachan Gurubachan Indian Institute of Technology Jodhpur India Co-Author(s):    Dr. Saurabh Verma and Dr. V. V. M. S. Chandramouli

Abstract: The first Heisenberg group $\mathbb{H}=\mathbb{R}^3$ is a homogeneous group equipped with the non-Euclidean metric $d_{\mathbb{H}}((x,y,t),(x`,y`,t`))=[((x`-x)^2+(y`-y)^2)^2+(t`-t+2(xy`-x`y))^2]^{\frac{1}{4}}$. Corresponding to the notorious fractal Sierpi\`nski gasket $(SG)$, one can get a fractal in $\mathbb{H}$ which is entitled as lifted Sierpi\`nski gasket $(\widetilde{SG})$. Motivated from the previous work on Heisenberg group and $\alpha$-fractal functions on $SG$ (Publ. Mat. 47 (2003), no. 1, 237-259, Proc. London Math. Soc. (3) 91 (2005), no. 1, 153-183, J. Math. Anal. Appl. 487 (2020), no. 2, 124036, 16 pp.), we first construct the class of $\alpha$-fractal interpolation functions (FIFs) in numerous spaces on $\widetilde{SG}$. Next, we define the $\alpha$-fractal operator with respect to the class of $\alpha$-FIFs constructed in numerous spaces, and also discuss the several properties of the same operators. We conclude the talk by providing the glimpse of the dimension of the graphs of constructed $\alpha$-FIFs.

Holder vs Dini in Transfer Operators

Yunping Jiang The City University of New York, Queens College and Graduate Center USA Co-Author(s):

Abstract: Ruelle transfer operators are central to thermodynamic formalism, providing critical insights into phenomena such as the existence and uniqueness of Gibbs measures, calculations of pressures and Hausdorff dimensions, and estimates of correlation decay rates. These operators exhibit distinct behaviors across different function spaces. This discussion will explore the contrast between transfer operators on Holder and Dini function spaces. I will discuss how the spectral gap can be used to analyze the exponential decay of correlations in expanding dynamical systems with Holder potentials. However, the spectral gap vanishes when the dynamical systems are not fully expanding or the potentials are not Holder. In collaboration with Yuan-Ling Ye, we established an optimal quasi-spectral gap condition for studying transfer operators in weakly expanding systems with Dini potentials. This condition enables us to derive precise estimates for the decay rate of correlations.

Dynamics of a 2-D discrete neuron map: nodal and network

Ajay Kumar Indian Institute of Technology Jodhpur India Co-Author(s):    V.V.M.S. Chandramouli

Abstract: We introduce a novel 2-D discrete neuron map, denoted by map H(x,phi), formed by incorporating electromagnetic flux into a one-dimensional Chialvo map. Our exploration encompasses a comprehensive study of the dynamical aspects of the map, covering fixed points, bistability, various bifurcations, S-shape attractor, firing patterns, pathways leading to chaos, including the period-doubling to chaos and reverse period-doubling to chaos. We validate the results by employing various dynamical techniques (like Lyapunov exponent diagrams, phase portraits, calculating the correlation dimensions, and basins of attraction). Beyond single-neuron analysis, we extend our investigation to a neuron network governed by the map H(x, phi), specifically a ring-star network configuration. This broader examination reveals various dynamical states within the network, including synchronous, asynchronous, and chimera states. Finally, we present simulations exploring different coupling strengths to uncover diverse wavy patterns and clustered states within the network dynamics.

Some Results on Graph Induced Symbolic Systems

Puneet Sharma Indian Institute of Technology Jodhpur India Co-Author(s):    Prashant Kumar

Abstract: In this talk, we will discuss the non-emptiness problem and some dynamical notions for graph induced symbolic shifts. We derive sufficient conditions under which non-emptiness problem and existence of periodic points for the shift space can be guaranteed. We also investigate properties such as transitivity, directional transitivity, weak mixing, directional weak mixing and mixing for the shift space under discussion. We provide necessary and sufficient criteria to establish horizontal (vertical) transitivity of the underlying shift space. We also provide examples to establish the necessity of the conditions imposed.

Effect of bounded linear operators on dimension

Saurabh Verma Indian Institute of Information Technology Allahabad, Prayagraj India Co-Author(s):

Abstract: In this talk, we first recall a few Banach spaces, such as continuous function space, continuously differentiable function space, bounded variation space, H\older spaces, oscillation spaces, and convex-Lipschitz spaces defined on a compact interval of the real line. Then, we will discuss the effect of some bounded linear operators on fractal dimensions of the graphs of functions belonging to these spaces. We also shed light on the impact of some well-known positive operators on the dimension.

Ergodic theory on coded shift spaces

Christian Wolf CUNY Graduate Center USA Co-Author(s):    Tamara Kucherenko, Martin Schmoll

Abstract: In this talk we present results about ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of H\{o}lder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also discuss flexibility results for the entropy on the sequential and residual set. Finally, we present a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov.

An Innovative Implicit-Explicit Fitted Mesh Higher-Order Scheme for 2D Singularly Perturbed Semilinear Parabolic PDEs with Non-Homogeneous Boundary Conditions

Narendra Singh Yadav Indian Institute of Information Technology, Sri City, Chittoor India Co-Author(s):    Kaushik Mukherjee

Abstract: This research presents an advanced numerical technique designed for solving two-dimensional singularly perturbed semilinear parabolic convection-diffusion equations, characterized by time-dependent non-homogeneous boundary conditions. The proposed method is a combination of an implicit-explicit fitted mesh method (FMM) and a Richardson extrapolation approach. The temporal discretization is handled through an Alternating Direction Implicit-Explicit (ADI) Euler scheme, which ensures accurate handling of time-dependent boundary values. For spatial discretization, we employ a hybrid finite difference scheme on a non-uniform rectangular mesh, while the time domain is discretized using a uniform grid. To begin, the paper explores the stability and asymptotic behavior of the analytical solution for the nonlinear problem. Following this, the stability properties of the implicit-explicit method are examined, and the convergence of the numerical solution is established, showing that the method achieves uniform convergence with respect to the perturbation parameter $\varepsilon$. The Richardson extrapolation technique is applied specifically to the time variable to enhance the accuracy and order of convergence in the temporal dimension. The proposed method is validated through a series of numerical experiments that confirm the theoretical predictions regarding stability and convergence rates. These numerical results demonstrate the effectiveness of the method in handling complex boundary conditions and solving the nonlinear parabolic PDEs with high accuracy.