**Dumitru Baleanu**

Cankaya University, Turkey

**Fractional Calculus and AI: Theory and Applications**

###### Abstract

Fractional calculus is an important mathematical tool with applications in several fields of science and engineering. Besides, the artificial neural network is very important for machine learning and it is a strong tool for modelling complex phenomena. In this talk I will present some applications of combined fractional calculus and artificial neural networks for solving real world problems.

**Jose Balthazar**

Member of Academy of sciences (ACIESP), SP, Brazil

Universidade Estadual Paulista, Brazil

**What Does Nonideal Transportation Mechanisms in MACRO and MEMS Scales Mean? Present, Past, & Future Directions Considering Regular and Irregular Motions**

###### Abstract

On the actual demand of modern technology, in the modeling of such structures, it is unavoidable to consider nonlinearities of the basic equations. New phenomena in Dynamics as well as new approaches to older ones are expected to be discovered in the theoretical, numeric, and experimental investigations of those structures. It is also well known that the study of problems involving the coupling of several systems was widely explored, in the last years, essentially in function of the change of constructive characteristics of the machines and structures. Accordingly, oscillatory processes can be divided into the following types: free, forced, parametric and self-excited oscillations and we remarked that two or more oscillations can interact in the same oscillatory system. This fact is of important scientific and practical interest. In this way, some phenomena were observed in a composed dynamic system supporting structures and rotating machines, where was verified that the unbalancing of the rotating parts was the greatest causer of the vibrations.We also noted that a lot of oscillatory(vibrating) phenomena of real systems cannot be explained by and solved based on linear theory, and it is important to introducing nonlinear characteristics into the mathematical models of vibrating systems and to electro-mechanical systems The main difficulty in comparisons of linear systems mainly because of absence of validity of superposition principle. Every nonlinear vibrating system must be solved individually, and special methodology must be developed for each class of problems. Here, we will deal with a special class of nonlinear systems called non-ideal systems (NIS).

** **

This lecture deals with the non-linear __Electro-mechanical Systems__ analysis of a block ( portal)foundation structure for an unbalanced rotating machine, with limited power, leading to the interaction between the motor and the structure(RNIS).This aspect is often not considered during the usual design practice, although all real motors are, in this sense, non-ideal energy sources coupled to the structure to describe the most real behavior for these structures. Nonlinear dynamic coupling between energy sources and structural response should not be ignored in real engineering problems, as real motors (agitators and so on) have limited output power. Therefore, numerical, and analytical methods are applied for such analyzes to better understand the non-linear dynamics of these systems. However, the common phenomena for these systems are the Sommerfeld effect, the occurrence of the saturation phenomenon and so on.

So, this lecture addresses the nonlinear dynamical analysis of a block foundation structure for an unbalanced rotating machine, with limited power supply, leading to interaction between the motor and the structure. This aspect is often not considered during usual design practice, although all real motors are, in this sense, non-ideal power sources. The considered mathematical model considers this system as non-ideal, subjected to the Sommerfeld effect, which may manifest close to foundation/machine’s resonances, with possible jumps from lower to higher frequency rotation regimes, no intermediate stable steady states in between. We also remarked that an additional property of (RNIS) with two degrees of from can be observed when the adopted model was calibrated, a 2:1 internal resonance occurs between the frequencies of the second mode and the first mode of vibration. It is intended to demonstrate that the energy pumped into the system via the second mode, leads to the well know saturation phenomenon (the energy balance to the first mode, not directly excited, which starts to develop wide amplitudes, potentially dangerous and not predicted in theory.

Other discussions addressed are the specific properties for various models considering transport mechanisms at MACRO and MEMS scales and their applications in engineering and science. The direction of future investigation is given by including a proposed mathematical model of energy harvesting, including non-linearities in the piezoelectric coupling and a non-ideal excitation force. So, we show through numerical simulations the non-linear dynamic responses that the collected power was influenced by the vibrations of the structure, as well as by the influence of the piezoelectric coupling.

Another interesting aspect is that the increase in voltage in the DC motor led the system to produce power responses, which showed the high energy orbits in the resonance region. However, for regions above the resonance region, the Sommerfeld effect occurs, and the dynamics has a chaotic behavior. Thus, the energy captured over time decreases due to energy losses due to the interaction between the energy source and the structure. Keeping the captured energy constant over time is essential to enable the use of energy harvesting systems in real applications. To achieve this goal, we apply a control technique to stabilize the chaotic system in a stable periodic orbit, that is, the control kept the system in a stable condition. Thus, the aim of this project is to present a state of the art to better understand non-ideal systems using numerical, analytical methods and some experiments.

By other hand, Timoshenko ‘s Beam Bending Model and the Sommerfeld Effect.Timoshenko’s beam bending model is an alternative to Euler-Bernoulli’s bending theory. In the latter, only the effects of the bending moment are considered, ignoring the acting shear forces. In Timoshenko’s theory, the effects of shear are also considered.If a beam is supporting a motor with limited power supply, which can lead to the so-called Sommerfeld Effect (stagnation of rotations at resonance and possible jumps), the Timoshenko hypothesis, although more complex, is preferable when the beam is relatively short in relation to the section dimensions, in which case the shear effect is appreciable. This is the case with short consoles .For a long beam, the Euler Bernoulli hypothesis is acceptable and simpler to apply.The adoption of any of these two hypotheses will of course reflect on the free vibration frequencies computation, specially upon the resonance frequency for which the Sommerfeld Effect may happen. As a more refined theory, Timoshenko’s beam bending theory should be a better approximation to the real behavior of such a system and is particularly better for high frequency excitations.

Finally, we say that this lecture is an overview of the literature dealing with the main properties of non-ideal vibrating systems. The analytical and numerical methods applied for analyzing such systems are shown. Practical examples of non-ideal systems are considered. The most common phenomenon for the systems is discussed (Sommerfeld effect, saturation phenomenon occurrence, and so on) . The specific properties for various models are also discussed, considering Transportation mechanisms in MACRO and MEMS scales and their applications to engineering and sciences. The direction of the future investigation is given. So, the aims of this lecture are to present a state of the art to better understand non ideal systems using numerical and some experiments.

KEYWORDS: Non ideal motor(RNIS), Mathematical Modeling, Nonlinear Systems, Control System, Chaotic Behavior.

**Pierpaolo Belardinelli**Polytechnic University of Marche, Italy

**Unconventional Stochastic Switching Events in Nonlinear Graphene Resonators**

###### Abstract

Nonlinear graphene resonators have recently been shown to be sensitive to thermal fluctuations close to room temperature. In this study, we investigate the nonlinear dynamics of a graphene nanodrum in relation to noise and while subject to an additional driving component far below the resonance frequency. The exceptional sensitivity of the graphene resonator leads to deviations from the conventional stochastic resonance scenario. These deviations are associated with alterations in the stable solutions and the paths of escape from metastable attractors. A theoretical model is employed to describe the intermittent occurrence of stochastic switching along with motion within a single quasiperiodic attractor. Our work thus reveals the significant effects of slow modulation on a nonlinear graphene resonator, which have important implications for sensing applications.

**Yoshihiro Deguchi**

Tokushima University, Japan

**Digital Twin Advanced Control of Industrial Processes Integrating Laser Diagnostics and CFD**

###### Abstract

In this study, a digital twin advanced control method integrating laser diagnostics and computational fluid dynamics (CFD) was developed for the digital transformation (DX) of industrial processes. Computed Tomography-Tunable diode laser absorption spectroscopy(CT-TDLAS) and Laser Induced Breakdown spectroscopy(LIBS) monitoring systems were developed to monitor the concentration and temperature of industrial processes. CT-TDLAS is based on the CT method using absorption spectra of molecules such as H_{2}O, CO_{2}, CO, O_{2}, NH_{3}, and hydrocarbons and allows real-time measurement of temperature and concentration distributions in two and three dimensions. LIBS is an analytical detection technique based on atomic emission spectroscopy for determining elemental composition. The integration of laser diagnostics and CFD enables the prediction of nonlinear phenomena such as combustion, and the feasibility of digital twin advanced control has been developed and demonstrated using a CH_{4}-NH_{3} swirl burner.

**Mark Edelman**

Yeshiva University, USA

Courant Institute, NYU, USA

**Generalized Fractional Multidimensional Maps**

###### Abstract

Formerly introduced generalized fractional maps are generalizations of the regular area/volume preserving maps. The examples of non-area preserving maps that can’t be generalized using the formerly proposed technique include the classical Hénon and Lozi maps. We propose a natural extension of the notion of the generalized fractional map which includes non-area preserving maps and investigate the general properties of these maps.

**Xilin Fu**Shandong Normal University, China

**Discontinuous Dynamics for Impulsive Differential Systems****with the State-dependent Impulses**

###### Abstract

Up to now, most of the research on impulsive differential systems focuses on the case with fixed time pulse, and rarely considers the case with arbitrary time pulse. The basic characteristic of the impulsive differential system with arbitrary time pulse is that its pulse depends on the state, which leads to “pulsation phenomenon” in the pulse surface. In this talk, from the viewpoint of discontinuous dynamical system, the pulse system which depends on state pulse is regarded as a discontinuous system bounded by a surface represented by pulse surface function. We use the measurement method of the discontinuous boundary flow transformation of discontinuous dynamical systems, build the mapping structure of the flow collision pulse surface by means of the characteristics of pulse mapping and pulse surface function，and the corresponding complex dynamical results of these systems are obtained.

This work is finacially supported by National Natural Science Foundation of CHINA No. 11971275.

**Igor Franović**Institute of Physics Belgrade, Serbia

**New insights on the impact of ketogenic diet on seizure dynamics from the next-generation neural mass models**

###### Abstract

**Celso Grebogi**Member of the World Academy of Sciences

University of Aberdeen, UK

**Recent Achievements In Studying Ecological Networks**

###### Abstract

Long-term predictions constitute a fundamental challenge in ecology, epidemiology, and climate science. Reliable forecasting is difficult because of sensitive dependence on initial conditions (the hallmark of chaos), noise, and incomplete data. Another obstacle to reliable prediction of ecosystems is a phenomenon known as “regime shift”, where any conclusions or estimates based on the observations made before the regime shift become irrelevant after the shift. The timing of the regime shift is difficult to predict and the problem of identifying early warning signals remains largely open. Sudden regimes shift often results in a population collapse, extinction of species and biodiversity loss, making it an important issue for nature conservation and ecosystem management. Focusing on the regime shift or the tipping-point dynamics in ecological mutualistic networks of pollinators and plants constructed from empirical data, I will examine the phenomena of noise-induced collapse and noise-induced recovery, aiming at understanding the interplay between transients and stochasticity. I will discuss control strategies that delay the extinction and advances the recovery by controlling the decay rate of pollinators in a stochastic mutualistic complex network, whose control strategies are affected by Gaussian environmental and state-dependent demographic noises, all having implications to managing high-dimensional ecological systems. Since in recent years, the concept of multilayer networks has also been adopted in ecology, I will also look at the influence of the topological structure on the control effect due to multiplexity. This is a basic notion in complex multilayer networks, where a subset of nodes belongs simultaneously to different network layers. I will argue that multiplexity also arises in multilayer ecological networks supported by mutualism and, more importantly, it has the fundamental benefits to sustaining the whole networked system and keeping it in a healthy state by delaying, often significantly, the occurrence of a catastrophic tipping point that would otherwise lead to extinction on a massive scale.

———–

**Control of tipping points in stochastic mutualistic complex networks**, Y. Meng and C. Grebogi, Chaos 31, 023118(1-9) (2021)

**Sudden regime shifts after apparent stasis: Comment on long transients in ecology**, C. Grebogi, Physics of Life Reviews 32, 41 (2020)

**Predicting tipping points in mutualistic networks through dimension reduction**, J. Jiang, Z.-G. Huang, T.P. Seager, W. Lin, C. Grebogi, A. Hastings, and Y.-C. Lai, PNAS (Proc. Nat. Acad. Sci.) 115, E639-E647 (2018)

**Noise-enabled species recovery in the aftermath of a tipping point**, Y. Meng, J. Jiang, C. Grebogi, and Y.-C. Lai, Phys. Rev. E 101, 012206 (2020)

**Tipping point and noise-induced transients in ecological networks**, Y. Meng, Y.-C. Lai, and C. Grebogi, J. Royal Soc. Interface 17, 20200645 (2020)

**The fundamental benefits of multiplexity in ecological networks**, Y. Meng, Y.-C. Lai, and C. Grebogi, J. R. Soc. Interface 19, 20220438 (2022).

**Markus Kirkilionis**University of Warwick, UK

**Multi-Scale Stochastic Modelling Framework For Complex Systems**

###### Abstract

**Jürgen Kurths**Member of the Academia Europaea

Humboldt University Berlin, Germany

**Forecasting extreme events related to tipping elements in the climate system**

###### Abstract

Tipping elements are components of the Earth system that may shift abruptly and irreversibly from one state to another at specific thresholds. An important aspect is how different tipping points are interrelated and how do the corresponding teleconnections are changing due to forcing. Here, we propose a climate network approach to analyse the global impacts of a prominent tipping element, the Amazon Rainforest Area (ARA). We find that the ARA exhibits strong correlations with regions such as the Tibetan Plateau (TP) and West Antarctic ice sheet. Models show that the identified teleconnection propagation path between the ARA and the TP is rather robust under climate change. We further uncover that various climate extremes between the ARA and the TP are synchronized and it is discussed how they can be forecasted.

**Nikolay V. Kuznetsov**

Member of Russian Academy of ScienceSt.Petersburg State University, Russia

**Global stability boundary and hidden attractors****in the phase-locked loops models**

###### Abstract

Phase-locked loops (PLLs) are classical nonlinear control systems for phase and frequency synchronization in electrical circuits. Design and analysis of frequency control circuits is a challenging task relevant to many applications: satellite navigation, telecommunications equipment, distributed computer architectures to mention just a few. From a broad perspective, their synthesis and analysis fall under the framework of standard topics in control engineering like signal tracking, linear and global stability. Meanwhile, some of ubiquitous and actively used circuits are largely inspired by implementability issues and approaches of practical control engineering so that their true capacities and limitations still await fully disclosing via a rigorous analysis.

A key engineering problem in nonlinear analysis of phase-locked loops (PLLs) is the determination of the pull-in range for the input signal frequency depending on physical implementation parameters of the system. For given physical implementation parameters and input signal frequency corresponding to the pull-in range, the state of the system with any initial data is attracted to a stationary set, and the system is called globally stable.

The boundary of global stability in the parameter space of the system is the boundary of the closure of the set of parameters for which the system is not globally stable (in the phase space, there are trajectories that do not tend to the stationary set). A point of the global stability boundary is called hidden if for its certain neighborhood in the parameter space the loss of global stability is caused only by global bifurcations of the birth of hidden oscillations for which the basin of attraction in the phase space is not connected with unstable equilibria; otherwise, the point is called explicit (trivial). Explicit parts of boundaries can be revealed by applying well-developed methods for analysis of local bifurcations and numerical analysis of self-excited oscillations in a neighborhood of unstable points of the stationary set. Methods for revealing hidden parts of the global stability boundary require nonlocal analysis, including analysis of global bifurcations, and such methods are developed in the theory of hidden oscillations.

- V. Kuznetsov, M.Y. Lobachev, M.V. Yuldashev, R.V. Yuldashev, The Egan problem on the pull-in range of type 2 PLLs, IEEE Transactions on Circuits and Systems-II: Express Briefs, 68(4), 2021, 1467-1471 (http://dx.doi.org/10.1109/TCSII.2020.3038075)
- Kuznetsov N.V., Lobachev M.Y., Yuldashev M.V., Yuldashev R.V., Tavazoei M.S. The Gardner problem on the lock-in range of second-order type 2 phase-locked loops, IEEE Transactions on Automatic Control, 2023, 68(12), 7436-7450 (https://dx.doi.org/10.1109/TAC.2023.3277896)
- KuznetsovV., Matveev A.S., Yuldashev M.V., Yuldashev R.V., Nonlinear analysis of charge-pump phase-locked loop: the hold-in and pull-in ranges, IEEE Transactions on Circuits and Systems I: Regular Papers, 68(10), 2021, 4049-4061 (https://doi.org/10.1109/TCSI.2021.3101529)
- Kuznetsov N.V., Lobachev Y., Mokaev T.N., Hidden Boundary of Global Stability in a Counterexample to the Kapranov Conjecture on the Pull-In Range, Doklady Mathematics, 108(1), 2023, 300-308.
- V. Kuznetsov, T.N. Mokaev, V.I. Ponomarenko, E.P. Seleznev, N.V. Stankevich, L. Chua, Hidden attractors in Chua circuit: mathematical theory meets physical experiments, Nonlinear Dynamics, 111, 2023, 5859–5887 (https://doi.org/10.1007/s11071-022-08078-y)

**Edson Denis Leonel**São Paulo State University, Rio Claro, Brazil

**Unraveling the Mysteries: Exploring a Second-Order Phase Transition in Chaotic Systems**

###### Abstract

**Zhihui Li**China Aerodynamics Research and Development Center, China

**Numerical forecast and computational modeling of nonlinear mechanical behavior of structural response induced by strong aerodynamic thermal environment during re-entry of large-scale spacecraft**

###### Abstract

How to solve the hypersonic aerothermodynamics around large-scale spacecraft on expiration of service and nonlinear mechanical behavior in deformation failure and disintegration of metal (alloy) truss structure during falling crashing process from outer space to earth, is the key basis to resolve the numerical forecast for the falling area of the reentry crash and flight path after the completion of the spacecraft mission. To study aerodynamics of spacecraft reentry covering various flow regimes, a Gas-Kinetic Unified Algorithm (GKUA) has been presented by computable modeling of the collision integral of the Boltzmann equation over tens of years. On this basis, the rotational and vibrational energy modes are considered as the independent variables of the gas molecular velocity distribution function, a kind of Boltzmann model equation involving in internal energy excitation is presented by decomposing the collision term of the Boltzmann equation into elastic and inelastic collision terms. Then, the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions by developing the discrete velocity ordinate method and numerical quadrature technique. The unified algorithm of the Boltzmann model equation involving thermodynamics non-equilibrium effect is presented for the whole range of flow regimes. The gas-kinetic massive parallel computing strategy is developed to solve the hypersonic aerothermodynamics with the processor cores of 500~45,000 at least 80% parallel efficiency. To validate the accuracy of the GKUA, the hypersonic flows are simulated including the reentry Tiangong-1 spacecraft shape with the wide range of Knudsen numbers of 220~0.00005 by the comparison of the related results from the DSMC and N-S coupled methods, and the low-density tunnel experiment etc. The multi-body flow problems including two and three side-by-side cylinders and irregular bodies are simulated with different gap ratio in highly rarefied to near-continuum flow regimes to verify the accuracy and reliability of the GKUA in solving the multi-body aerothermodynamics for spacecraft falling disintegration, see Fig.1. To develop the computational modeling from the hypersonic flow field to the solid structure, a coupling mathematical model of transient heat conduction equation and thermo-elastic dynamic equation of materials is presented for falling and crashing problem during the unconventional reentry of un-controlling spacecraft, and then the finite-element algorithm of dynamic thermal-force coupling response is presented under strong aero-thermodynamic environment. The coupling computational technique of reentry aerodynamic environment and structural thermal response is developed, and the nonlinear mechanical behavior of structural response deformation of vertical plate and Tiangong-type vehicle was integrated to compute and verify under reentry aerodynamic environment. Then, the forecasting analysis platform of end-of-life large-scale spacecraft flying track is established on the basis of ballistic computation combined with reentry aerothermodynamics and deformation failure /disintegration. The accuracy and reliability of the large-scale parallel computing strategy and finite element algorithm are verified to reveal the thermal response deformation, and then, The integrated simulation platform has been applied to the uncontrolled falling of Tiangong-1 target spaceship, and the controlled reentry disintegration of the Tiangong-2 space laboratory. Figure 2 shows the forecast of flight trajectory and falling area of disintegrated wreckage and debris from the uncontrolled Tiangong-1 spacecraft falling with the comparison of the USA monitoring results afterwards from the map calibration of NASA’s post official website (www.space-Track.org) announcement in good agreement and compatibility. It is indicated from the falling reentry forecast that the uncontrolled Tiangong-1 will be disintegrated firstly at the range of 110-105km, secondly at 100-95km, specially, the main bearing-cone platform and trajectory controlling engines will be disintegrated at 83km-56km and so on. The present numerical forecasting platform obtained the scope of falling area distribution of longitudinal length 1200km and lateral width 100km from the first disintegration to debris falling to the ground. These results on the multiple disintegration, falling area distribution and trajectory calculation of coupled aerothermodynamics for the uncontrolled Tiangong-1 spacecraft affirm the accuracy and reliability of the unified modeling and typical computation of structural response and hypersonic aerothermodynamics for falling disintegration along ballistic trajectory with different flying height, Mach numbers covering various flow regimes from outer space to earth.

**Keywords:** Numerical forecast of nonlinear mechanical behavior of structural response; Tiangong-1 target vehicle; Aerothermodynamics covering various flow regimes; Boltzmann model equation in Thermodynamic non-equilibrium effect; Gas-kinetic unified algorithm; finite element algorithm of structural response deformation / destruction

**Wei Lin**Fudan University, China

**Machine learning techniques meet complex dynamical systems: A few recent advances**

###### Abstract

In the era of big data, machine learning techniques, as the core of AI research, have been widely and efficiently used to deal with the observational data produced by the complex dynamical systems. In spite of the tremendous progress, the development of lightweight machine learning techniques are limited but highly anticipated for achieving systems reconstruction and evolution prediction. Here, we report a few advances in this direction, integrating dynamical system theory with causality analytics and lightweight recurrent neural networks. We demonstrate the efficacy of the developed approaches using several representative systems and their produced data of physical and biological significance. We also offer some perspective on the further development in this interdisciplinary research domain.

**Albert C.J. Luo**Southern Illinois University, Edwardsville, USA

**Limit Cycles and Homoclinic Networks in 2-dimensional Polynomial Systems**

###### Abstract

To be updated.

**Elbert E. N. Macau**Federal University of Sao Paulo – UNIFESP, Sao Paulo, Brazil

**Phase synchronization in a sparse network of randomly connected neurons under the effect of Poissonian spike inputs**

###### Abstract

We pesent results about the emergence of phase synchronization in a network of randomly connected neurons by chemical synapses. The study uses the classic Hodgkin-Huxley model to simulate the neuronal dynamics under the action of a train of Poissonian spikes. In such a scenario, we observed the emergence of irregular spikes for a specific range of conductances, and also that the phase synchronization of the neurons is reached when the external current is strong enough to induce spiking activity but without overcoming the coupling current. Conversely, if the external current assumes very high values, then an opposite effect is observed, i.e. the prevention of the network synchronization. We explain such behaviors considering different mechanisms involved in the system, such as incoherence, minimization of currents, and stochastic effects from the Poisso- nian spikes. Furthermore, we present some numerical simulations where the stimulation of only a fraction of neurons, for instance, can induce phase synchronization in the non-stimulated fraction of the network, besides cases in which for larger coupling values it is possible to propagate the spiking activity in the network when considering stimulation over only one neuron.

**Sergey Meleshko**Suranaree University of Technology,

Thailand

**Group Analysis of the Stationary Magnetogasdynamics Equations in Lagrangian Coordinates**

###### Abstract

**Fuhong Min**Nanjing Normal University, China

**Complex Firing Behavior and Synchronization Analysis of Heterogeneous Neural Network**

###### Abstract

In this report, the heterogeneous neural network model coupled with the Hindmarsh-Rose and FitzHugh-Nagumo is investigated with the help of the discrete mapping method. Diverse bifurcation behaviors and discharge phenomena are first found by varying the coupling strength. Stable orbits, and even unstable orbits are presented to describe the firing patterns through phase plane. Additionally, discrete nodes in the phase, time-histories diagrams of the membrane potentials and deviations of the membrane potentials are employed to analyze the correctness of the discrete mapping method. Subsequently, the synchronous and asynchronous behaviors depending on the coupling strength are successively revealed with changing the parameters. Finally, the physical feasibility of the heterogeneous neural network by implementing field-programmable gate array (FPGA) circuit is validated.

**Maaita Jamal-Odysseas**

Aristotle University of Thessaloniki, Greece

**Dynamical behavior of systems with positive maximal Lyapunov Characteristic Exponent very close to zero**

###### Abstract

In recent years, many scientists have introduced systems with interesting dynamic behaviors.

As is known, the maximal Lyapunov Characteristic Exponent indicates whether the system behaves regularly or chaotically. If the system has a positive maximal Lyapunov Characteristic Exponent, it is chaotic; otherwise, it is regular.

Also, more than one Lyapunov Characteristic Exponent indicates the system has a hyperchaotic behavior.

In many papers, systems with Lyapunov Characteristic Exponents very close to zero are called chaotic or hyperchaotic. The fact that the Lyapunov Characteristic Exponents are numerically calculated raises the following question: When we consider the Lyapunov Characteristic Exponent positive rather than a numerical mistake? In other words, how close to zero must the value of the Lyapunov Characteristic Exponent be to say that it equals zero?

In this talk, we will discuss the dynamic behavior of several dynamic systems with Lyapunov Characteristic Exponent very close to zero, investigate their behavior, and set criteria and procedures to check the positivity of the Lyapunov Characteristic Exponent.

**Pawel Olejnik**

Lodz University of Technology, Poland

**The Application of a Neural Network Based on a Physical Model of Rotational Tribological Contact for Determining Asymmetric Friction Law**

###### Abstract

**Lev Ostrovsky**University of Colorado, Boulder, USA

**Damping and Amplification of Turbulence**

**in the Ocean: Theory and Measurements**

###### Abstract

**Vakhtang Putkaradze**

University of Alberta, Canada

**Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries**

###### Abstract

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable.

We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special integrals of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff’s equations for an underwater vehicle (SE(3) group), and others.

Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (NTU Singapore), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

Keywords: Machine learning, Hamiltonian systems, Lie-Poisson brackets

**Victor Shrira**

Keele University, UK

**Self-induced transparency of weakly dispersive nonlinear waves in non-uniform media: the dispersive shock mechanism**

###### Abstract

We argue that effect of self-induced transparency via the dispersive shock mechanism is universal for weakly dispersive weakly nonlinear waves in media with slow inhomogeneities. We borrow the term ` self-induced transparency’ from nonlinear optics where phenomenology is similar: waves of large initial amplitude pass through the media, while for the smaller amplitude waves the media is less transparent. The underpinning physics is entirely different. In nonlinear optics quantum interference allows for the propagation of high intensity light through an otherwise opaque medium. In the dispersive shock wave mechanism we consider here, weakly nonlinear waves are propagating through an inhomogeneity: in linear setting waves are effectively reflected by an inhomogeneity of the media when the wavelength of the incident wave is comparable or exceed the scale of inhomogeneity, in contrast, a weakly nonlinear wave tend to disintegrate forming a dispersive shock wave (DSW) which is very effective in transferring energy into much shorter scales. These shorter waves pass through the inhomogeneity with a negligible reflection.

As the basic illustrative physical example we examine in detail long surface waves in fluid propagating over topography. Dispersive shock waves (DSW) are a salient feature of long waves often observed in tidal bores and tsunami/meteotsunami contexts. The shoreline hazard from tsunamis and meteotsunamis critically depends on the fraction of incoming energy flux transmitted across the shallow nearshore shelf. By considering nonlinear dynamics of waves over variable depth within the framework of the Boussinesq equations we show that the transmitted energy flux fraction can strongly depend on the initial amplitude of the incoming wave and the distance it travels. We show that this is an order one effect. The same logics holds for all weakly dispersive weakly nonlinear wave fields.

**Carla Pinto**

Instituto Superior de Engenharia do Porto, Portugal

**Resampling methods are used for inverse uncertainty quantification in stochastic systems**

###### Abstract

In this talk we describe the use of the Bayesian bootstrap for inferring posterior distributions of parameters in stochastic differential equations with parametric uncertainty. The proposed method applied resampling residuals from deterministic least-squares optimization with Dirichlet weights, simplifying repeated deterministic calibrations. Additionally, we incorporate the principle of maximum entropy for densities, to handle parameters not optimized deterministically. We present one case study on HIV evolution patterns, in a patient under clinical follow-up.

**Miguel AF Sanjuan**Member of Spanish Royal Academy of Sciences

^{}Rey Juan Carlos University in Madrid, Spain

**Symphony of the Uncertainty in Three Movements**

###### Abstract

In this presentation, akin to a symphony in three movements, I will provide an overview of various concepts related to uncertainty and unpredictability in nonlinear dynamics. I will start from the topological concept of indecomposable continua to Wada basins. Then, I will continue with the Wada basins of the Hénon-Heiles Hamiltonian system, to finally culminate with the concept of basin entropy. Additionally, I will discuss recent advancements in using basin entropy for classifying basins, as well as for exploring bifurcations.

[1] J. Aguirre, R.L. Viana, and M. A. F. Sanjuán. Fractal structures in nonlinear dynamics. Reviews of Modern Physics 81, 333-386 (2009)

[2] A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin, and M. A. F. Sanjuán. Basin entropy: a new tool to analyze uncertainty in dynamical systems. Scientific Reports 6, 31416 (2016)

[3] A. Daza, A. Wagemakers, and M. A. F. Sanjuán. Unpredictability and basin entropy. Europhysics Letters 141, 43001 (2023)

[4] A. Daza, A. Wagemakers, and M. A. F. Sanjuán. Classifying basins of attraction using the basin entropy. Chaos Solit. Fractals. 159, 112112 (2022).

[5] A. Wagemakers, A. Daza, and M. A. F. Sanjuán. Using the basin entropy to explore bifurcations. Chaos Solit. Fractals. 175, 113963 (2023).

**Michael Small**The University of Western Australia, Australia

**Dynamics of Machine Learning**

###### Abstract

What is old is new again. Machine learning can be understood as attempts to apply data driven techniques to uncover underlying deterministic dynamics, or to approximate it through stochastic methods. In this talk I will describe three approaches to understand machine learning from the perspective of dynamical systems. First, recurrent neural networks will be shown to perform an embedding of time series data in the sense of Takens’ theorem. That is, the internal state of the neural network is diffeomorphic to the underlying (presumed determinsitic) dynamical system. Second, while generative Artificial Intelligence achieves sentient-like performance through a carefully orchestrated random walk we will see how this can be construed as a stochastic dynamical system represented by a walk on a graph (or Markov chain). Thirdly, I will describe the application of learning techniques to estimate the state of a network dynamical system from observation of the node dynamics. Along the way, the utility of these methods will be demonstrated with application to industrial maintenance, music and detection of ventricular fibrillation.**Bio**: Michael is the CSIRO-UWA Chair of Complex Systems, Director of the UWA Data Institute, and a Professor of Applied Mathematics at UWA. His research group works on applying the mathematical theories of Chaos and Complexity to data-driven problems. Current applications of his group include disease modelling, predicting propensity for self-harm and suicidal ideation, traffic modelling, geological discovery under cover, and predictive maintenance. Prof. Small is Chief Investigator of the ARC Research Hub for Transforming Energy Infrastructure Through Digital Engineering, and the ARC Training Centre for Transforming Maintenance through Data Science. He is a Former Future Fellow and current Deputy Editor in Chief of the journal Chaos. In 2022 was awarded the V. Afraimovich Award of the Nonlinear Science Society for contributions to complex systems and nonlinear dynamics.

**Haris Skokos**University of Cape Town, South Africa

**Quantifying chaos using Lagrangian descriptors**

###### Abstract

References:

1. Madrid J. J., Mancho A. M., “Distinguished trajectories in time dependent vector fields”Chaos, 19, 013111 (2009).

2. Mancho A. M., Wiggins S., Curbelo J., Mendoza C., “Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems” Communications in Nonlinear Science and Numerical Simulation, 18, 3530—3557 (2013).

3. Hillebrand M., Zimper S., Ngapasare A., Katsanikas M., Wiggins S., Skokos Ch., “Quantifying chaos using Lagrangian descriptors” Chaos, 32, 123122 (2022).

4. Zimper S., Ngapasare A., Hillebrand M., Katsanikas M., Wiggins S., Skokos Ch., “Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map,” Physica D, 453, 133833 (2023).

5. Skokos Ch., “Alignment indices: A new, simple method for determining the ordered or chaotic nature of orbits,” J. Phys. A, 34, 10029—10043 (2001).

6. Skokos Ch., Manos T., “The Smaller (SALI) and the Generalized (GALI) alignment indices: Efficient methods of chaos detection,” Lect. Notes Phys., 915, 129—181 (2016).

**Alexander Solynin**Texas Tech University, USA

**Quadratic Differentials in Analysis and Theoretical Physics**

###### Abstract

To be updated.

**Yury Stepanyants**University of Southern Queensland, Australia

**Advanced theory of solitons, lumps, and ripplons ****in the cylindrical Kadomtsev−Petviashvili equation**

###### Abstract

We revise soliton and lump solutions described by the cylindrical Kadomtsev–Petviashvili (cKP) equation and construct new exact solutions relevant to the comparison with physical observations. In the first part of this study, we consider basically axisymmetric waves described by the cylindrical Kortweg–de Vries equation and analyse approximate and exact solutions to this equation. Then, we consider the stability of the axisymmetric solitons with respect to the azimuthal perturbations and suggest a criterion of soliton instability. The results of our numerical modelling confirm the suggested criterion and reveal lump emergence in the course of the development of the modulation instability of ring solitons in the unstable case. In the next part of this study, we present exact solutions to the cKP equation describing weakly nonlinear waves in media with positive dispersion subject to the modulation instability of solitons with respect to small azimuthal perturbations. By means of Zakharov–Shabat method, we derive exact solutions that describe two-dimensional solitary waves (lumps), lump chains, and their interactions. One of the obtained solutions describes the modulation instability of outgoing ring solitons and their disintegration onto a number of lumps. We also derive solutions describing decaying lumps and lump chains of a complex spatial structure – ripplons. Then, we study normal and anomalous (resonant) interactions of lump chains with each other and with ring solitons. The results obtained agree with the numerical data presented in the first part of this study.

In collaboration with

W. Hu^{1}, Q. Guo^{2}, and Zh. Zhang^{2}

^{1}School of Physics and Optoelectronic Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China;^{2}Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, P. R. China.

**Pierre E. Sullivan**University of Toronto, Canada

**Reduced Order Modeling of Flow Over a Low Reynolds Number Airfoil**

###### Abstract

A single dielectric-barrier discharge plasma actuator is an active flow control (AFC) device that imparts momentum to the fluid through ion acceleration using electromagnetic forces and has been used to suppress flow separation. This presentation studies flow over an airfoil and how adding an SDBD actuator influences flow characteristics through numerical modeling. Using the spectral proper orthogonal decomposition and large-eddy simulation (LES), flow instabilities in the wake region are analyzed at their different temporal and spatial scales. These results are compared to resolvent analysis/ The objective of this study is to explore the viability of spectral proper orthogonal decomposition (SPOD) and resolvent analysis for separation control and correlate the decomposed flow modes to the different modes of actuation.

**Jianqiao Sun**University of California, Merced, USA

**Neural Network-based Subspace Harmonic Expansion for Obtaining Highly Accurate Periodic Solutions of Nonlinear Dynamic Systems**

###### Abstract

The periodic responses of a nonlinear dynamic system are crucial for vibration analysis in engineering. The typical harmonic balance method and its variants can derive the analytical expression of the periodic steady-state solutions by solving the Fourier coefficients of harmonic terms. However, it is confined to low-order approximations limited by the ability of symbolic operations. To achieve highly accurate periodic solutions, high-order harmonic components have to be retained in the truncated approximation. Still, solving a large number of harmonic coefficients could lead to the curse of dimensionality or frequency aliasing. In this work, we conduct the task of solving harmonic coefficients in the neural network framework, where the neurons construct a base function space consisting of harmonic components of the response. The base space is discretized into grids to identify the covering set of harmonic coefficients. Therefore, determining harmonic coefficients are regarded as discovering invariant sets in the discrete base space. Based on the idea of subspace expansion, the covering set can be found first in a low-dimensional subspace, and then progressively extended to higher dimensional subspaces as more neurons are activated. Furthermore, most low-order coefficients (weights) that remain invariant in the lower level of grids can be froze and inactivated during the backpropagation when conducting subspace expansion. This approach significantly reduces the computation costs in high-dimensional optimization. Notably, spatial discretization serves as the basic framework for guiding machine learning for setting appropriate network parameters, rather than approximating point-wise results. Additionally, how to capture multiple physical solutions by means of error-tolerance of discretized space are also discussed. As a result, we can simultaneously obtain multiple high-order (>200) periodic solutions at once, previously deemed out-of-reach, for several challenging examples.

In collaboration with

Zigang Li^{1}, Wang Yan^{2}, Miao Li^{1}

1 College of Sciences, Xi’an University of Science and Technology, Xi’an, Shaanxi 710054, China

2 School of Mechanical Engineering, Xi’an University of Science and Technology, Xi’an, Shaanxi 710054, China

*Speakers*

Dr. Jian-Qiao Sun earned a BS degree in Solid Mechanics from Huazhong University of Science and Technology in Wuhan, China in 1982, a MS and a PhD in Mechanical Engineering from University of California at Berkeley in 1984 and 1988. He worked for Lord Corporation at their Corporate R&D Center in Cary, North Carolina. In 1994, Dr. Sun joined the faculty in the department of Mechanical Engineering at the University of Delaware as an Assistant Professor, was promoted to Associate Professor in 1998 and to Professor in 2003. He joined University of California at Merced in 2007, and is currently a professor of the Department of Mechanical Engineering in School of Engineering. Besides many other editorial experiences, he is the Editor-in-Chief of International Journal of Dynamics and Control published by Springer.

His research interests include stochastic non-linear dynamics and control, cell mapping methods, multi-objective optimization, intelligent control systems and high-density piezoelectric energy harvesting from highway traffic.

Dr. Zigang Li earned his Ph.D. from Xi’an Jiaotong University in 2016 and completed his postdoctoral research at UC Merced in the United States in 2019. He joined Xi’an University of Science and Technology at 2016, and is serving as an Associate Professor and doctoral supervisor in the field of Mechanics. His research interests include data-driven and machine learning algorithms, nonlinear global dynamics and control, and rotor dynamics. He has published over 30 academic papers in many international journals and conferences.

**Edgardo Ugalde**

Instituto de Física – UASLP, Mexico

**An Elementary Approach to Subdiffusion**

###### Abstract

We consider a basic one-dimensional model that allows us to obtain a diversity of diffusive regimes whose speed depends on the moments of a per-site trapping time. These models are discrete subordinated random walks, closely related to the continuous time random walks widely studied in the literature. The models we consider lend themselves to a detailed elementary treatment, based on the study of recurrence relation for the time-t dispersion of the process, making it possible to study deviations from normality due to finite time effects.

**Luis Vázquez**Complutense University of Madrid, Spain

**Title: To be updated**

###### Abstract

To be updated.

**Dimitri Volchenkov**Texas Tech University, USA

**Thermodynamic Analysis of Network Dynamics: Insights from Very Long Walks**

###### Abstract

We discuss the thermodynamic behaviors of very long walks on finite, connected transportation graphs, considering random modifications and transportation noise. We highlight how statistical mechanics can quantify structural dynamics in networks, especially in complex systems subject to random changes. Nodes with low centrality are prone to future alterations. The analysis extends to engineered systems, emphasizing their non-random, robust structures. It discusses how traffic perturbations affect network dynamics, particularly in urban settings. Structural defects reshape mobility patterns, with entropic forces influencing network evolution. Finally, we delve into the statistical properties of long paths, emphasizing the Fermi-Dirac distribution and its implications for network modifications.

**Guo-Cheng Wu**

Chongqing University of Posts and Telecommunications, China

**Data-driven and Deep Learning of Fractional Difference Equations**

###### Abstract

This talk introduces several key problems in deep learning of fractional difference equations. A general fractional calculus is revisited and the function space is provided. By use of the time scale theory, Hadamard and Exponential discrete fractional calculus are defined and propositions are obtained. Finally, parameter estimation of fractional difference equations is investigated and its perspective is shown in deep learning.

**Jiazhong Zhang**

Xi’an Jiaotong University, China

**LCSs-based Fine Functional Structures in Unsteady Fluid Flows**

###### Abstract

From viewpoint of dynamic system and topological physics, unsteady flow is one dissipative dynamic system, the initial study shows that there exist intrinsic spatiotemporal structures in complex unsteady flows, and their topology properties have significant influences on the flow performance. The Hyperbolic, Elliptic and Parabolic Lagrangian Coherent Structures (LCSs), with invariant spatiotemporal properties in a period, are introduced and developed to describe and analyze the complex flow structures in unsteady flows, and some numerical methods following nonlinear dynamics are given in capturing LCSs. Further, some fine functional structures, like energy sink, targeted energy transfer, are shown and studied in aeroacoustics and others. As the results, the methods based on complete intrinsic Lagrangian Coherent Structures could describe and analyze the flow structures and dynamics of complex unsteady flows quantitatively, and further control the flow with one new and accurate method.

**Weimou Zheng**Institute of Theoretical Physics, Chinese Academy of Sciences, China

**Applied Symbolic Dynamics: from 1D to 2D**

###### Abstract

To be updated.

**Zhigang Zheng**Huaqiao University, China

**Self-organization Dynamics of Chiral Swarmalators**

###### Abstract

Self-organized swarmings of interacting chiral oscillators (swarmalators) are studied.

The mechanisms, forms, and manifestations of the emergence of various collective swarmed states are explored at both microscopic and macroscopic levels. These studies shed light on both theoretical understandings and potential applications of collective behaviors in complex systems.

This work is finacially supported by National Natural Science Foundation of CHINA No. 12375031.