**Christos Volos**

###### A Dream that has Come True: Chaos from Nonlinear Circuits with Physical Memristors

__Abstract:__ The concept of memristor has been presented in a seminal paper by Prof. Chua in 1971. Since then, only few works had been reported in literature for a long time. The main reason for this was that the new proposed circuit element was only a theoretical concept. Therefore, until 2008, memristor had received little attention.

However, that year, researchers in Hewlett-Packard published an article in the Nature in which a physical model of memristor was presented. Since then, a great number of articles have been published presenting different models of memristors, design techniques and fabricated materials. As it is known, one of the most important features of memristor is the nonlinear relationship between current and voltage. So, in the last decade, researchers, who work in the field of nonlinear circuits, have the “dream” to

use a real memristor, as a nonlinear element in a new or other reported nonlinear circuit, in order to experimentally investigate chaos. With this intention, the last couple of years some first works in this direction have been presented. Therefore, in this talk the first nonlinear circuits, in which its nonlinear elements have been replaced with commercially available memristors, will be discussed. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors,

one-scroll and double-scroll chaotic attractors, which have been experimentally observed, will be presented. Finally, some thoughts for future works, such as the use of physical memristors in real-world applications will be discussed.

**Cristina Muresan**

###### Fractional order control: step towards industry use

__Abstract: __Fractional calculus and its use in the design of

fractional order controllers has received a lot of attention during the past

years. However, most of this interest is focused on research and development,

with very few applications in the industrial sector. This is partly due to the

engineer’s sceptisism regarding fractional order controllers and to the

researcher’s failure of developing tuning and implementation methods that are

simple enough to be used in practical applications. Most of the processes in

the industry are controlled via the simple PID controller, so perhaps

developing automatic tools and devices that implement the generalized

fractional order PID controller would facilitate their use in such industrial

systems. In this talk, we explore the autotuning methods developed for

fractional order PIDs and the most efficient ways to implement them, both in

terms of simplicity and resource minimization. The end purpose would be to

see whether such an automatic tool for fractional order PIDs design would make

these controllers more appealing to practical applications.

**Dimitri Volchenkov**

Texas Tech University, Department of Mathematics and Statistics

###### Diagnostic Uncertainty in Quarentine

__Abstract:__ Diagnostic reliability is essential in epidemiology, in part because reliability is necessary for validity, not to say that patients are experiencing symptoms differently. We calculate the amounts of predictable and unpredictable information components in a three-state biased coin Susceptive – Infected – Immune, for the different configurations of transition bias and under random transition times between states (while in quarantine). We show that if vaccination does not guarantee immunity from a virus it worsens the diagnostic reliability. Randomness of transition times worsens the reliable diagnostic as well. The reliable diagnostic of patients may not be possible.

**Dmitry V. Kovalevsky**

Climate Service Center Germany (GERICS), Helmholtz-Zentrum Geesthacht

**Igor L. Bashmachnikov**

The Saint Petersburg State University

Nansen International Environmental and Remote Sensing Centre

**Genrikh V. Alekseev**

Arctic and Antarctic Research Institute

###### Nonlinear Dynamics of Open-Ocean Deep Convection

__Abstract:__ Open-ocean deep convection occurs when surface water penetrates down to several kilometers, as a result of the gravitational instability of the upper water column which leads to a vigorous overturning. Observed only in a few localized sites of the World Ocean, this phenomenon is an important element of the Global ocean conveyor belt and, therefore, of the global climate system. The complexity of physics of deep convection and a broad spectrum of spatial and temporal scales of various physical processes involved lead to a broad spectrum of modelling approaches applied to the analysis of nonlinear dynamics of this phenomenon. In this presentation, we focus on two analytically tractable models: (1) a model of a development and a shutdown of a deep convective ‘chimney’ (a vertically homogenized water column in the stratified surrounding ocean) and (2) a model of water exchange of a localized deep convective site with the neighboring ocean. This second model is a generalization of Whitehead’s ‘tank model’. Dependent on the values of non-dimensional parameters, the model yields qualitatively different flow regimes and allows for multiple steady states. The modelling results are tested against observations in the Greenland Sea. In spite of their simplicity,the models adequately describe many features of water dynamics during real deep convective events.

I.B. acknowledges a support during this study of SPbSU project 75295423.

**Duarte Valério**IDMEC, Instituto Superior Técnico, University of Lisbon

### Nonlinear fractional dynamics of cancer growth

__Abstract:__ Tumour growth can be modelled using nonlinear equations. More generally, the evolution of the tumour in both space and time can normally be described using systems of nonlinear partial differential equations. Diffusion processes are involved, and diffusion can be anomalous. In the case of tumours, anomalous diffusion can be expected because the medium is not homogenous, and the very presence of the tumour increases this effect. Anomalous diffusion is itself the result of a nonlinear relationship, and has as a consequence the appearance of fractional order derivatives in the differential equations that describe its dynamics.

Additionally, diffusion phenomena related to the pharmacokinetics and pharmacodynamics of drugs used in tumour treatments can lead to fractional order derivatives. Fractional order dynamics have been found a useful tool to model the growth of cancer cells in bone tissue, and also to model the corresponding treatment. Tumour growth and metastasis phenomena in other organs have more recently profited from the same approach. An overview is presented of the different types of nonlinear models for this disease that benefit from the inclusion of fractional derivatives, of the results obtained, of the insights that such models allow, of

the relation with optimal control problems, and of the possibilities that a

fine-tuning of model parameters to fit data specific for a particular patient

opens for personalised medicine.

**Edgardo Ugalde **

Autonomous University of San Luis Potosí

###### Dominant vertices for piecewise dissipative maps on networks

__Abstract:__ In this talk, we consider a class of dynamical systems inspired by some systems of differential equations used to model gene regulatory networks. These systems consist of a network of units whose states evolve following dissipative dynamics that change at each time step. The choice of the local dynamics to follow at each time step depends piecewise constantly on the state of the neighboring units, which establishes a threshold-type interaction between the nodes of the network. Due to the dissipative character of the local dynamics and the interdependence encoded by the network, it is in principle possible to reduce the description of such a system. In previous work, we showed that the orbit in a well-chosen subset of nodes, which we call dominant vertices, determines the asymptotic behavior of the entire system. In this talk, we will present recent results concerning the role of the dominant nodes in the global dynamics, particularly with regard to dynamic complexity. Our results apply to state-discrete systems such as the Boolean networks and do not require the dynamics to be synchronous.

Reference: B. Luna and E. Ugalde, “Dominant vertices in regulatory network dynamics”, Physica D 237 (21) (2008) 2685-2695.

**Efthymia Meletlidou**

###### Continuation of periodic orbits in perturbed integrable Hamiltonian systems (Hamiltonian and dissipative)

Abstract: A 2n dimensional integrable Hamiltonian system has orbits that are lying on n n dimensional tori. If the system is nondegenerate then on a dense set of them there exist periodic orbits.

If we add a small perturbation then generically according to the Poincare-Birchoff theorem there are a finite set of periodic orbits for each resonant torus (which carries periodic orbits of the unperturbed system). For a specific system we use Poincare ‘s theorem to find which of the orbits are continued in the perturbed system. When there are degenerate tori in the

system we use a theorem of our own to find the continuation of periodic orbits for specific systems. Finally for a case of a nonlinear oscillator with a linear oscillator with a dissipation and a degenerate perturbation we find hidden limit cycles.

**Fuhong Min **

Nanjing Normal University

###### Discontinuous dynamic analysis of memristive neuronal circuit with a picewise flux-controlled memristor

Abstract: In this paper, the discontinuous dynamical behaviors of two memristive neuronal circuit with piecewise memristor are studied. The necessary and sufficient conditions for motion switchability at the boundary are investigated through the theory of discontinuous dynamical systems. The various motions through the boundary with the distinct initial conditions and system parameters are demonstrated by the parameter maps and coexisting bifurcation diagrams. The coexistence of an example system under different initial conditions is illustrated by attraction basins, and the corresponding trajectories in phase plane. The periodic and chaotic motions with different mapping structures are analyzed for a better understanding of the switching mechanism.

**Jian-Qiao Sun**

University of California

###### Identification of Nonlinear Mechanical Systems with Time Delay

__Abstract__: In this talk, we present some recent studies on identification of nonlinear mechanical systems with time delay. With the available measured response of the system and a rough knowledge about the nature of the mechanical system, we propose to use a general polynomial to describe the nonlinearity of the damping and restoring forces. To this end, the number of terms in the polynomial is minimized with the sparse optimization algorithm. The proposed method of this study combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We also apply the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. The Taylor expansion of the time delay term is used to make the time delay explicit for estimation. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

**Jiazong Zhang (Xian)**

Xi’an Jiaotong University

###### Analysis of Singular Phenomena in Complex Unsteady Flow with Dynamic System Approach

The results show that the extraordinary flow performances, which are closely related to the “intrinsic properties” of the unsteady flow field, can be induced by unsteady singular flow, such as nonlinear vortex lift and unsteady super-high lift in the flow around the airfoil. However, due to the complexity of unsteady flow, the quantitative description and analytical approach of the “intrinsic properties” of the unsteady flow have always been a difficult problem in fluid dynamics and there is no a specified method, and such situation constrains the in-depth studying and utilization of the nature of singular phenomena in unsteady flows.

Indeed, there is still a lack of rational explanations and laws of nonlinear phenomena in complex unsteady flows. One of the main reasons is the absence of method to quantitatively describe and analyze the intrinsic structures (modes and basis) and dynamic features of complex unsteady flows. Therefore, the way to capture and quantitatively describe the intrinsic structures in complex unsteady flows and to analyze its dynamical characteristics, including both mass transport and mixture, is of great challenge and academic significance in fluid community.

Following differentiable dynamic system and theories in nonlinear dynamics, flow fields are one typical type of infinite-dimensional dynamic systems in which there exist some intrinsic flow structures, that is, flow modes and gene with manifold invariance. Through several years study, it is proved that Lagrangian Coherent Structures (LCSs) in the flow field possess such property. As a result, they can be used to represent the steady basic skeletons hidden in velocity field.

Based on present LCSs, Networks as well as Coherent sets are introduced to further develop an approach to describe the topology of LCSs in quantitative way.

Finally, as a verification of the above discussions, some examples relevant to unsteady flows are presented to illustrate the feasibility and advantages of the analytic method of unsteady singular flows based on dynamic system theories.

**Maaita Jamal-Odysseas**

Aristotle University of Tessaloniki

###### Nonlinear Targeted Energy Transfer in systems of coupled linear and nonlinear damped oscillators

__Abstract:__ The one-way directed energy transfer from an initial system (donor) to a final nonlinear system (receptor) is called nonlinear Targeted Energy Transfer. The great significance of nonlinear TET lies in the fact that systems in which Nonlinear TET occur present a form of self-tuning and can transfer energy over a wide variety of frequencies (resonances). This makes nonlinear TET particularly suitable in practical applications where it is necessary to extract energy from multiple oscillation ways.

We will discuss nonlinear TET in coupled linear and nonlinear oscillators with different damping types, make a comparison and describe the differences and the similarities between them

**Mark Bala**

Texas A&M University Engineering

###### Exponentially Convergent Nonlinear Estimation of the Quantum Density Operator for a Quantum Mechanical Statistical System Using Metric Projection Operators

Abstract: The fundamental element of quantum statistical mechanics is the quantum density operator. Once determined this operator reveals the statistics of observables in a quantum process. We will present our research on the estimation and approximation of the quantum density operator from the Liouville-Von Neumann master equation. It is critical to retain all the properties of quantum operators during approximation and estimation; so that at any stopping point during the estimation process the result will be a true quantum density operator. The set S of all quantum density operators ρ on the Hilbert Space X of trace class operators contains all self-adjoint positive semidefinite operators with trace one. This set is contained inside the unit ball and is a closed convex set. The metric projection P_s(x) satisfies d(x,P_s(x))=d(x,S)=inf d(x,z), x ∈ X . It exists and is unique for each x ∈ X , and is a monotone, Lipschitz continuous operator on X. We create a linear state estimator for Quantum Statistical Systems described by the Liouville-Von Neumann Master equation on X . The output ρˆ(t) does not remain in S for all t>0 even though it converges exponentially to the true ρ(t) of the quantum statistical system. However, we show that Pρˆ(t ) does remain in S for all t>0 and also converges exponentially to the true ρ( )t as well. Thus, the metric projection Pρˆ( t) makes the state estimation process nonlinear but creates an invariant set of S for this nonlinear state estimator. This provides convergent estimates of ensemble averages of Observables for the Quantum Statistical System under consideration.

**Panayotis G. Kevrekidis**

University of Massachusetts Amherst

###### Some Vignettes of Nonlinear Waves in Granular Crystals: From Modeling and Analysis to Computations and Experiments

__Abstract:__In this talk, we will provide an overview of some results in the setting of granular crystals, consisting of beads interacting through Hertzian contacts. In 1d we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will discuss the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria analogous to the famous Vakhitov-Kolokolov criterion in the NLS model. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with recent experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

**Raoul Nigmatullin**

Kazan National Research Technical University

###### The detection of self similar/branching processes in complex biological systems: Analysis of the temporal evolution of impedance measurements in Tulsi (Holy Basil) leaves "Ocimumtenuiflorum"

__Abstract:__ In this paper the proposed theoretical model describing the branching systems in the intermediate range of frequencies was applied in the first time for description of the living Tulsi (Hole Basil) leaf evolution. The proposed fitting function that follows from the applied model allows to describe completely with high accuracy (the fitting error was less than 0.1%) the whole stages of the temporal evolution of the leaf during 6 days of the impedance measurements. The fitting parameters that follow from the model enables to select at least three stages. The first stage can be defined as the conserved vital leaf activity. It occupies approximately 1 day. Then during next three days the complex impedance exhibits a quasi-chaotic behavior, when the behavior of the imaginary part of the complex impedance becomes non-monotone. It can be defined as the transient stage when the measured leaf “enters” to the “dying” stage. The last two days correspond to the withering leaf stage. The discovery of the branching processes in complex biological systems allows to understand deeper the transfer charge processes in the intermediate range of available frequencies scales. Besides, this model implies that new fractal element with complex conjugated power-law exponents really exists in nature. Its experimental confirmation can play a key role in explanation of a wide class of branching processes in complex systems and enrich also the modern theory of the fractional calculus.

**Renat Sibatov **

Ulyanovsk State University

###### Fractional nonlinear dynamics in electrochemical power sources with nanostructured electrodes

__Abstract:__ This talk reviews applications of nonlinear fractional equations to mathematical modeling of charge dynamics in supercapacitors (SC) and lithium-ion batteries (LIB) with nanostructured electrodes. Mechanisms peculiar to ion transport in electrode nanoparticles, and electrolytes filled porous electrodes are discussed. For these mechanisms, we highlight features leading to fractional dynamics. Within the framework of the phase field model based on the time-fractional Cahn-Hilliard equation, the model of diffusion formation of an electric double layer in an electrolyte filling pores of different geometry and size is revised and the frequency dependences of the Warburg impedance are generalized. We studied charge transfer in an electrode, where conductivity is provided by pure ionic or combined ion-electron charge transfer. The driving force is the chemical potential of the ions, which is described in terms of the phase field, which allows us to avoid additional calculation of the activity coefficient. Electrochemical responses of SC and LIB are considered for various experimental techniques. The links between microscopic kinetic models and equivalent circuits containing fractional impedances are discussed.

We acknowledge support from the Russian Science Foundation (project 19-71-10063).

**Santanu Saha Ray **

National Institute of Technology in Rourkela

###### Analytical and Numerical Methods in Modelling Riesz Fractional Reaction-Diffusion-Mechanics System and Riesz Fractional Nonlinear Schrödinger Equation in Quantum Mechanic

__Abstract:__ A finite difference method with implicit scheme for the Riesz fractional reaction-diffusion equation has been proposed by utilizing the fractional centered difference for approximating the Riesz derivative, and consequently it leads to an implicit scheme which is proved to be convergent and unconditionally stable. Also a novel analytical approximate method has been dealt with namely optimal homotopy asymptotic method (OHAM) to investigate the solution of Riesz fractional reaction-diffusion equation. The numerical solutions obtained by proposed implicit finite difference method have been compared with the solutions of OHAM and also with the exact solutions. The comparative study of the results establishes the accuracy and efficiency of the techniques in solving Riesz fractional reaction-diffusion equation. The proposed OHAM renders a simple and robust way for the controllability and adjustment of the convergence region and is applicable to solve Riesz fractional reaction-diffusion equation.

Also, a nonlinear Schrödinger equation with the Riesz fractional derivative has been considered. This equation has been solved by two reliable methods in order to investigate the accuracy of the solutions. In the implicit finite difference numerical scheme, the fractional centered difference is utilized to approximate the Riesz fractional derivative. Also a novel modified optimal homotopy asymptotic method with Fourier transform (MOHAM-FT) has been proposed to compute the approximate solution of Riesz fractional nonlinear Schrödinger equation. Further the numerical solutions obtained by proposed implicit finite difference method, have been compared with that obtained by MOHAM-FT to exhibit the effectiveness of the suggested methods. Finally, the obtained solutions have been presented graphically to justify the efficiency of the methods.

**Sverre Holm **

University of Oslo

###### Searching for the physics of fractional derivative models

__Abstract:__Fractional derivative models fit many kinds of data across applications, but do these models point to some deeper physics, or are they just a compact phenomenological description? Here, some proposals for underlying models will be discussed. The first is in terms of multiple relaxation processes with a distribution of relaxation times that follows a power-law distribution. Such a distribution hints at some fractal properties in the underlying medium. A second way is a mechanical model in the form of a rheopectic material modeled with a time-varying non-Newtonian viscosity or its electrical counterpart, a time-varying inductor or capacitor. Under the right conditions, the step or impulse responses of such models is a power-law function just like that of a fractional derivative model. Third, a medium with a fractal distribution of scatterers can also give rise to power law attenuation and dispersion. The presentation builds on and develops ideas from Chaps. 7-9 in Holm, Waves with Power-Law Attenuation, 2019.

**Xavier Leonicini **

Aix-Marseille University

###### Motion of a charged particle in a magnetic filed, from stationnary Maxwell-Vlasov solution to Hamiltonian chaos and the “destruction” of the magnetic moment

__Abstract:__ Starting from a given passive particle equilibrium particle cylindrical

profiles, a self-consistent stationary conditions of the Maxwell-Vlasov

equation at equilibrium. Furthermore the full motion of charged particles in a magnetic toroidal configuration is analyzed. The destruction of a separatrix

and the emergence of Hamiltonian chaos, is displayed. In these regions the

magnetic moment, shows large fluctuations and is not a conserved adiabatic

quantity. This leads to questions the validity of the hypothesis behind

gyrokinetics.