**Mahmoud Abdel-Aty**

Sohag University, Egypt

**Nonlinear Dynamics of Quantum Entanglement**

###### Abstract

In this work, we will examine in a proof-of-concept experiment a new type of quantum-inspired protocol based on the idea of nonlinear dynamics of quantum entanglement. We discuss various measures of bipartite and tripartite entanglement in the context of two and three level atoms. The quantum entanglement is discussed for different systems. For the three-level systems various measures of tripartite entanglement are explored. The significant result is that a sudden death and sudden birth of entanglement can be controlled through the system parameters.

**Mark Edelman**Yeshiva University, USA

**Stability of discrete fractional systems and lifespan of living species**

###### Abstract

Numerical simulations demonstrate that discrete fractional systems are

robust with respect to random perturbations. The distribution of times of the stable evolution before the break of unstable fractional (with power-law memory) systems under random perturbations coincides with the observed lifespan distribution of living species.

**Jordan Hristov**University of Chemical Technology and Metallurgy, Bulgaria

**FRACTIONAL MODEL FOR REAL WORLD PHENOMENA: ****Basic principles in construction, causality and fractional operator applications**

###### Abstract

The lecture addresses the correct approaches in constructions of fractional models of physical phenomena by operators with power-law and non-singular memories. The main line of the presentation follows basic thermodynamic prinples, causality conditions and causal fractionalization of constitutive equations. The examples provided are from the fields of mechanics, fluid and solid rheology, heat and mass transfer, diffusion, electrity, etc.

The main idea of this talk is to show that constructions of fractional models should be correctly done by following basic laws and last but not least, to demostrate that some clasical and correctly developed models, may serve as missleanding examples when new types of memory kernels are used. This especially addresses the formal fractionalization by replacement of integer-order operators in clasical models and already known fractional models with power-law memories when non-singular kernels are applied.

**Amin Jajarmi **University of Bojnord, Iran

**A new approach on the modelling and control of complex biological systems**

###### Abstract

Recently, the new aspects of fractional calculus have been widely employed to investigate different features of many complex biological systems. In this direction, the fractional models help us to understand how the memory of the certain components of a system affects the progress of diseases as a whole, and therefore, it enables us to implement the memory effects into the evolution of considered system together with its environment. This kind of analysis is also important in order to improve the current medications and to explore new ways of quick, effective and low-cost treatments.

In this talk, we explore a recent development in the mathematical modelling of biological systems. The complex dynamics of an epidemic are investigated within the use of both classical and a new fractional framework. The obtained results are analyzed by the help of some simulations in a comparative way for both the integer- and fractional-order models. Finally, an efficient control scheme is designed for the purpose of intervention in an appropriate, effective way.

**Devendra Kumar**

JECRC University, India

**An efficient numerical scheme of a fractional order nonlinear discontinued problems**

###### Abstract

In this work, we suggest an efficient and user friendly numerical technique for solving a nonlinear fractional discontinued problem occurring in arising in nanotechnology. We combine of an analytical method and a new integral transform to construct the numerical scheme for solving the fractional order problem with different kind of memory effects. This inventive coupling makes the calculation very simple. The results derived with the aid of the proposed scheme reveals that the scheme is very efficient, accurate, flexible, easy to use and computationally very attractive for such kind of fractional order nonlinear models.

**Stefano Lenci**

Polytechnic University of Marche (UNIVPM), Italy

**Everything you wanted to know about 1:1 internal resonance but never dared to ask**

###### Abstract

Internal resonance is the mechanics by which an energy exchange can occur in the nonlinear regime between modes that are uncoupled in the linear framework. This coupling can be dangerous, if not properly considered, or useful, if adequately detected and even exploited.

Although internal resonances, and in particular the 1:1, have been largely studied in the past for various mechanical systems, *a systematic, general and comprehensive investigation is missing*. It is the goal of this work, which is aimed at investigating the general case with all quadratic and cubic nonlinearities.

A cornucopia of different possibilities is observed by varying the nonlinear stiffnesses, and a detailed analysis is performed. Both modes can be hardening, softening, one hardening and the other softening, modes can be in-phase or out-of-phase.

Furthermore, the existence of an extra path of periodic solutions, not ensuing from any natural (linear) frequency is observed, seemingly for the first time, and its existence has been confirmed numerically.

The proposed finding can be conveniently exploited to design a coupled oscillator with desired (nonlinear) properties.

**Xavier Leoncini**CPT, Aix-Marseille Université, France

**From particle dynamics in magnetic field to building stationary solution of the Maxwell-Vlasov equation**

###### Abstract

**Edson Denis Leonel **UNESP-Universidade Estadual Estadual Paulista, Brazil

**An investigation of chaotic diffusion**

###### Abstract

In this talk I discuss how chaotic orbits diffuse in the phase space for both dissipative and non dissipative 2-D mappings. For conservative cases the phase space is mixed and chaos is present in the system leading to a finite diffusion in one of the dynamical variables. For the dissipative case chaotic attractor is present in the phase space and the diffusion is limited. Indeed the diffusion is investigated by the analytical solution of the diffusion equation under certain boundary conditions as well as specific initial conditions. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.

**Changpin Li **Shanghai University, China

**Stability and decay of the solution to Hadamard-type fractional differential equation**

###### Abstract

Hadamard-type fractional differential equation, i.e., fractional differential equation with Hadamard derivative, is one kind of important fractional differential equations, which may have potential applications in mechanics and engineering, e.g., the fracture analysis, or both planar and three-dimensional elasticities. In this talk, we mainly present the stability, asymptotic stability of the static solution (i.e., equilibrium) to the Hadamard-type fractional differential equation. In the case of asymptotic stability, the decay rate of the solution is also determined.

**Jose Tenreiro Machado **ISEP-Institute of Engineering of Porto, Portugal

**Fractional calculus mission: To explore strange new worlds**

###### Abstract

Fractional Calculus (FC) started with the standard differential calculus but remained an obscure topic during almost three centuries. The present-day popularity of FC in the scientific arena, with a growing number of researchers and published papers, makes one forget that 20 years ago the topic was considered “exotic” and that a typical question was “FC, what is it useful for?”

We recall two remarkable foreseeing quotes about FC: “It will lead to a paradox, from which one day useful consequences will be drawn” (G. Leibniz, 1695) and “The fractional calculus is the calculus of the XXI century” (K. Nishimoto, 1991). Indeed, new advanced and emerging areas of application of the future of FC.

Present day popular directions of progress are the formulation of new operators, the “fractionalization” of integer models, the further development of known topics and the pursuit for new areas of application. The first two, namely the proposal for new definitions of fractional-order operators, or the fractionalization of some mathematical models, may represent critical adventures with possible misleading or even erroneous formulations. The third, that is, the in-depth study of some mathematical and computational fields, constitute a solid option, but its lack of ambition narrows considerably the scope of FC. The fourth option leads to exploring new applications, both with mathematical and computational tools, and represents a challenging strategy for the progress of FC.

Possible new directions of progress in FC may emerge in the fringe of classical science, or in the borders between two or more distinct areas. The lecture presents some uncommon ideas and topics, namely the application of computational and visualization methods for the analysis of data series and the characterization of complex phenomena.

**Vladimir Nekorkin**Institute of Applied Physics, RAS, Russia

**Dynamics of oscillatory adaptive networks**

###### Abstract

Many real-world complex systems can be described as networks, i.e., sets of nodes connected by links. Usually, the pattern of links (structure or topology) is considered as a result of a process independent of nodes’ intrinsic states. However, for a large variety of cases, the nodes as well as the links exhibit dynamics that can shape the structure, i.e., one deals with an adaptive network.

We report the results of studying the dynamics of a network of phase oscillators (Kuramoto-type system) and a network of coupled Stuart-Landau oscillators. The nodes of the networks are coupled by adaptive links (coupling strength depends on nodes’ states) while the topology of the networks is either one-layer or two-layer. We show that in such networks there exist phenomena of self-organized emergence of hierarchical structures and hierarchical transitions. We consider chimera states and phenomena of their synchronization. And we talk about new, the third type of chaos – the so-called mixed dynamics, which is characterized by the fundamental inseparability of dissipative and conservative behavior; and some other phenomena.

**Raoul Nigmatullin**Kazan National Research Technical University (KNRTU-KAI), Russia

**What kind of “hidden” information can be extracted from usual “noise”?**

###### Abstract

In this abstract the author wants to prove that a trendless sequence (TLS) (usually determined as a noise”) can be used as an additional source of information. This additional information can be extracted from random noise with the help of 3D-DGIs (discrete geometrical invariants) method that allows to reduce 3*N* random data points to 13 parameters composed from the combination of integer moments and their intercorrelations up to the fourth order inclusive. Actually, they form a “universal” 13-feature space for comparison of one random sequence with another one. Comparison of these parameters associated with different noise tracks allows to use this set of parameters for calibration and other purposes associated with “standard”/reference equipment. It is similar to ides used by the Hibbs, when the partition function transforms the 3*N* degrees of freedom associated with the movement of micro-particles and are described of the initial Hamiltonian to a finite set of thermodynamic parameters. A similar operation can be realized with the help of the proposed 3D-DGIs method. Actually, a general platform for comparison of different random sequences/functions is created. Thirteen parameters enable to transform a triangle matrix (*N* × *M*) (*N*-number of rows, *M*-number of columns) representing initial measurements to the reduced matrix of the form *M*×*Pr*. The parameter *Pr* =13 includes in itself the following combinations: (<x>,<y>,<z> -three centers of gravity, R_{1,2,3} – compact combination of the correlations of the third order, *A*_{11}, *A*_{22}, *A*_{33}, *A*_{12}, *A*_{13}, *A*_{23} -six correlations of the second order and *I*_{4} – invariant that includes the compact combination of the 4-th order correlations). One can show that the following reduction of the matrices *M*×*Pr* is possible also. These new possibilities give a researcher a new “universal” and very sensitive tool for reduction, comparison and further analysis of different TLS(s) and random functions with each other.

**Carla M.A. Pinto **University of Porto, Portugal

**Epidemiological models: usefulness and predictions in the era of COVID-19**

###### Abstract

In the era of COVID-19, people turn more and more to mathematical models, to gain insight and predict the course of epidemics. Politics turn to mathematicians, epidemiologists, for guidance on what should be the best control practices to avoid a disaster in terms of humans’ lives. The World is facing an unprecedent pandemic, with severe consequences in terms of loss of lives and economically.

With the utmost ideas in mind, in this talk, we will focus on the applicability of mathematical models of infectious diseases. We go from the usual and simple Susceptible-Infectious (SI) model to more realistic ones, which include variable transmission rate, treatment, intervention policies, non-integer order derivatives, amongst others. Some examples will be given for each model.

**Lev A. Ostrovsky**

University of Colorado Boulder, Boulder, Colorado, USA

**Modeling of complex two-dimensional patterns of interacting solitons**

###### Abstract

The two-dimensional, oblique interaction of solitary waves can form various complex patterns that were observed, in particular, for the internal and surface waves in the ocean and in the laboratory installations. This phenomenon is still waiting for a comprehensive theoretical analysis. We suggest a relatively simple kinematic approach to the description of interaction between plane solitons forming steadily moving structures. This approach is applicable to both integrable and non-integrable two-dimensional models possessing a soliton solution. It allows obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. With the help of this approach, one can determine the speed and direction of motion of two-soliton patterns (including resonant soliton triads) and find the reference frames where the patterns are stationary. The suggested approach is validated by comparison with the exact two-soliton solutions of the integrable Kadomtsev–Petviashvili (KP2) equation. Expanding the analysis with using an asymptotic theory allows determining the spatial shift of soliton fronts due to the oblique interaction. In the KP2 case, the phase shift derived from the asymptotic method completely coincides with what follows from the exact solution. The developed theory is applied to the available results of observations of the internal and surface waves in the ocean.

**Minvydas Ragulskis**Kaunas University of Technology, Lithuania

**Clocking convergence of discrete nonlinear maps (including fractional maps)**

###### Abstract

Discrete nonlinear maps have been extensively studied for more than five decades since the introduction of the logistic map as one of the first examples of a deterministic system exhibiting chaotic behavior. Algorithms for clocking the asymptotic and non-asymptotic convergence of non-invertible and completely invertible maps will be discussed in this talk. A computational technique based on the visualization of the algebraic complexity of transient processes will be employed for that purpose. Temporary stabilization of unstable orbits in non-invertible maps will be demonstrated and discussed. We will show that the dynamics of the ractional difference logistic map is similar to the behavior of the extended invertible logistic map in the neighborhood of unstable orbits. This counter-intuitive result will provide a new insight into the transient processes of fractional nonlinear maps.

**Hansong Tang **The City College of New York, USA

**Coupling of partial differential equations to simulate flow problems**

###### Abstract

In recent years, computer simulation of fluid flows sees a quick transition from solving a single partial differential equation (PDE) (or a single system of PDEs) into solving multiple PDEs coupled with each other (or coupled systems of PDEs). In the past, such simulation focuses on individual physical phenomena, which tend to be described by a PDE (or a system of PDEs), and now it has become a trend to simulate multiple phenomena depicted by multiple PDEs (or multiple systems of PDEs), presenting a multiscale/multiphysics simulations. Along with the development of numerical methods in coupling PDEs in flow simulation, this presentation will go over typical methods and applications as well as a few examples of current research, and it covers flow problems in various backgrounds (e.g., aerospace eng., environmental sciences, and ocean engineering).

**Vitaly Volpert**

CNRS, France

**Nonlocal and delay reaction-diffusion equations in mathematical immunology**

###### Abstract

Conventional models in mathematical immunology consist of ordinary or delay differential equations for the concentrations of different cells participating in the immune response and for the concentration of pathogen. Their spatial distribution in the tissue or cell culture, or their dependence on the genotype is described by reaction-diffusion equations with time delay characterizing clonal expansion of lymphocytes and with nonlocal terms taking into account cross reaction in the immune response. In this presentation we will study some mathematical properties of such models and their biomedical applications.

**Guo-Cheng Wu**

Neijiang Normal University, China

**Recurrent neural networks with short memory: A fractional calculus approach**

###### Abstract

Fractional derivative holds memory effects that have extensively implemented in dynamic systems and modeling. However, it is a challenging work to consider the discrete analogy. We use the fractional calculus on a time scale to define fractional discrete-time systems.Then we propose new recurrent neural network models, and the short memory approach is used to decrease the computational cost. Finally, the performance is shown in comparison with that of the classical recurrent neural network.