Dynamics

A graph-theoretical approach to the analysis of motion and rest in many-body systems is developed. Point bodies are represented as vertices of a complete bi-colored graph, termed the motion–rest graph (MRG). Two vertices are connected by a rust-colored edge when the corresponding bodies are at rest relative to each other; that is, when their mutual distance remains constant in time, bodies moving relative to each other are connected by a cyan edge. It is shown that the logical structure of the relation “to be at rest relative to each other” determines the combinatorial structure of the graph. For one-dimensional motion in classical mechanics and special relativity, this relation is reflexive, symmetric, and transitive, and therefore defines an equivalence relation. As a result, rust edges form disjoint complete cliques corresponding to rest-clusters, and the MRG becomes a semi-transitive complete bi-colored graph that is completely determined by the partition of the bodies into equivalence classes. It is proven that any such graph on five vertices necessarily contains a monochromatic triangle. For two- and three-dimensional motion, the transitivity of relative rest generally fails because constant mutual distance does not imply an equality of velocities in the presence of rotational degrees of freedom. In this case, the MRG is non-transitive, and the Ramsey threshold becomes the classical value R(3,3) = 6. The approach is extended to mixed sets containing moving bodies and reference points, including the center of mass of the system. Generalizations to general relativity and quantum mechanics are also discussed. In general relativity, transitivity of relative rest is generically lost because global rigid congruences do not generally exist. In quantum mechanics, exact transitivity survives only at the level of idealized delocalized eigenstates, whereas for physically realizable localized states, the notion of mutual rest becomes only approximate. The results demonstrate that the interplay between kinematics, logical properties of relational motion, and Ramsey-type combinatorial constraints gives rise to unavoidable ordered substructures in many-body systems. Full article

The results of the development and practical application of a comprehensive system for studying gear cutting processes are presented. The processes are traditional hobbing, modern power skiving, and radial-circular methods. Carrying out these processes is based on the gear teeth continuous generating method using complex kinematics. This complicates the analysis, description and modeling of the processes. The developed system provides for a logical sequence of step-by-step modeling and simulation of interrelated processes and phenomena accompanying gear processing. Reproducing volumetric chips and calculating their parameters provides the basis for determining deformation and contact processes, cutting forces, elastic deformations, machining accuracy and energy costs per operation. After establishing the operation to overcome friction and heat flows, the degree of heating and the temperature of the working surfaces are calculated to predict tool wear and its service life. Based on the parametric non-uniformity of the considered processes, the intensity of oscillations and vibrations of gear cutting machines is predicted, and their impact on the quality of gear surfaces and the accuracy of gears is determined. These approaches enable the study of such processes at the level of individual teeth and blades during cutting. They also allow gear cutting technology and cutting tools to be optimized according to the most important criteria and performance assessments. Full article

The Rössler system is a paradigmatic chaotic oscillator widely used to investigate synchronization phenomena. Existing studies on monovariate coupling almost exclusively rely on the x or y variables, while coupling through z is commonly regarded as ineffective. In this work, we report that complete synchronization through the z variable is indeed possible, provided that specific parameter values are chosen. We further consider a parameter regime in which the Rössler system exhibits multistability and show that synchronization via z-coupling occurs only when the dynamics evolve on a particular attractor. Although synchronization can be achieved, the admissible range of coupling strengths is very narrow as determined by the master stability function. For small networks, full connectivity is required, whereas larger networks can tolerate the removal of a limited number of links without losing synchronization. An analytical expression predicting the fraction of connections that must be preserved as a function of network size is derived and validated, revealing that a very high average degree is necessary. This effectively excludes common topologies such as small-world and scale-free networks. Numerical examples with up to 100 oscillators are presented, and potential challenges that may yield new insights are discussed. Full article

Simulated spiking neural networks have been explored for over a hundred years. Many of these networks are driven by biological considerations and an attempt to simulate brains, but others are used with little biological consideration. This paper gives some history of the development of spiking neural models, their use for modelling biological and cognitive phenomena, and for machine learning. It introduces the current state of the art in computational biological neuron and synapse modelling and plasticity. It introduces and reviews balanced spiking networks and their engineering applications. Spiking networks are also used for machine learning, with the hope that their implementation on neuromorphic hardware will bring energy and time savings. Similarly, neuromorphic hardware can enable massive parallelism, supporting larger spiking networks. The use of spiking nets for machine learning, both with biologically plausible models and without, is discussed, showing that effective models already exist. The paper concludes with some notes about implementing spiking nets and a discussion including open questions and future work. Full article

The growing need to safeguard sensitive data in various fields, including in relation to education, banking over the phone, private voice conferences, and the military, has grown as dependence on technology in daily life has increased. Encryption schemes based on chaotic systems are among the most commonly utilized approaches in the security field due to their high levels of safety and reliability. This study proposes a secure audio encryption framework based on the Chameleon chaotic algorithm implemented on a Xilinx ZedBoard Zynq-7000 FPGA. The system was designed using a fixed-point arithmetic format with 32-bit precision (eight integers; 24 fractional bits) with the Xilinx System Generator in MATLAB Simulink R2021b and verified using Vivado. The Chameleon Chaotic System, characterized by its transition from self-excited to hidden attractors through parameter variation, adds complexity to the system dynamics and strengthens the encryption algorithm. The Adaptive Feedback Control technique was applied to synchronize the signals. These methods enhance the security of audio data by ensuring robust and fast synchronization during transmission. The performance of the proposed system was assessed using correlation analysis, the mean squared error, histogram analysis, and audio spectrogram analysis. The system demonstrated strong encryption capabilities with low correlation values (−0.0033). In decryption, they achieved high fidelity with a correlation exceeding 0.999 in noise-free conditions and above 0.9933 under 20 dB AWGN. Adaptive Feedback Control showed superior decryption precision with lower MSEU and higher PSNR, confirming its effectiveness under noisy environments. Full article

This paper investigates the dynamics of a permanent magnet synchronous motor (PMSM) and controls its chaotic speed behavior using the synergetic control technique (SCT). The model includes electrical dynamics in the dq frame and mechanical speed dynamics, with a scalar parameter $\gamma$ capturing cross-coupling effects. The equilibrium structure and local stability properties of the PMSM are analyzed. For zero input voltages and zero load torque, the system exhibits a pitchfork-type bifurcation in the electrical–mechanical equilibrium as $\gamma$ crosses a critical value. Explicit expressions are derived for all equilibria, and their stability is characterized using eigenvalue analysis and the Routh–Hurwitz criterion, and a secondary loss of stability via a Hopf-type mechanism is identified. The case of nonzero input voltages with zero load torque is also discussed. Numerical simulations confirm the analytical results and highlight the parameter regions that admit stable operation. Bifurcation diagrams show the different PMSM behaviors as the parameter $\gamma$ varies. For a certain interval of $\gamma$, the PMSM speed undergoes chaotic oscillations. The SCT is introduced to control the chaos. Macro variables are chosen to design the SCT. The derived SCT is implemented to eliminate the chaotic speed. The controller provides good performance in suppressing the chaos. The controller is tested under sudden reference speed change where the controller gets the new reference speed accurately. It is also evaluated under sudden and sinusoidal load torque variations. Full article

We study the Cauchy problem for a loaded fractional integro-differential equation with a time-dependent diffusion coefficient. By reducing the problem to an equivalent Volterra integral equation of the second kind, we derive explicit analytical representations of solutions under appropriate regularity assumptions. The construction of the associated resolvent kernel allows us to establish existence and uniqueness results and to investigate the role of the fractional order and the loading term in the solution structure. Two illustrative examples are presented to demonstrate the applicability of the proposed approach. Full article

This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The Itô or Stratonovich stochastic differential equations with the Wiener component describe dynamic systems, and the manifold is implicitly defined by a differentiable function. A convenient implementation of the algorithm for forming invariant stochastic differential systems within symbolic computation environments characterizes the proposed method. It is based on determining a basis associated with a tangent hyperplane to the manifold. This article discusses the problem of basis degeneration and examines variants that allow for the simple construction of a basis that does not degenerate. Examples of invariant stochastic differential systems are given, and numerical simulations are performed for them. Full article

The dynamic behaviour of a column excited at the base, e.g., under an earthquake load, has been extensively studied. However, the column may also experience impact at the tip like a heavy-duty truck braking on a bridge. The caused base shear of the pier is very important. In this work, the dynamic behaviour, particularly the impact load from the tip to the base, was studied on two different composites: double basalt- and double flax fibre-reinforced polymer tube (DBFRP and DFFRP)-confined coconut fibre-reinforced concrete (CFRC). For each composite, two columns with a height of 1 m, an inner diameter of the outer tube of 100 mm, and an inner tube of 30 mm were fabricated. The column was fully fixed at the base and struck at the top with an impulse hammer. The base shear was calculated through an equivalent mass method using the acceleration at the tip. The results show that both DBFRP-CFRC and DFFRP-CFRC can dissipate a portion of the impact force, resulting in a reduction in force at the base of the specimens. The base shear of DFFRP-CFRC columns is larger and dissipates energy faster than that of DBFRP-CFRC columns. Full article

Empirical debates about a “crisis of trust” highlight long-lived pockets of high trust and deep distrust in institutions, as well as abrupt, shock-induced shifts between the two. We propose a probabilistic model in which such phenomena emerge endogenously from social learning on hierarchical networks. Starting from a discrete model on a directed acyclic graph, where each agent makes a binary adoption decision about a single assertion, we derive an effective influence kernel that maps individual priors to stationary adoption probabilities. A continuum limit along hierarchical depth yields a degenerate, non-conservative logistic–diffusion equation for the adoption probability $u(x,t)$, in which diffusion is modulated by $(1-u)$ and increases the integral of u rather than preserving it. To account for micro-level uncertainty, we perturb these dynamics by multiplicative Stratonovich noise with amplitude proportional to $u(1-u)$, strongest in internally polarised layers and vanishing at consensus. At the level of a single depth layer, Stratonovich–Itô conversion and Fokker–Planck analysis show that the noise induces an effective double-well potential with two robust stochastic phases, $u\approx 0$ and $u\approx 1$, corresponding to persistent distrust and trust. Coupled along depth, this local bistability and degenerate diffusion generate extended domains of trust and distrust separated by fronts, as well as rare, Kramers-type transitions between them. We also formulate the associated stochastic partial differential equation in Martin–Siggia–Rose–Janssen–De Dominicis form, providing a field-theoretic basis for future large-deviation and data-informed analyses of trust landscapes in hierarchical societies. Full article

Multi-revolution Lambert solvers are intended to find the elliptic transfer orbits that are traveled multiple times and connect two specified positions in prescribed time, under the assumption of considering natural (Keplerian) orbital motion in the presence of a single attracting body. This study proposes and tests a new, effective multi-revolution Lambert solver that employs the initial true anomaly, which identifies the initial position along the transfer ellipse, as the unknown variable. The related search interval is identified through closed-form expressions for upper and lower bounds. A simple numerical algorithm is developed and employed over the entire search interval to detect all Lambert solutions. The new multi-revolution solver proposed in this work is simple to understand and easy to implement and is successfully tested in several challenging scenarios (corresponding to some pathological cases reported in the recent scientific literature), as well as for the study of Earth–Mars interplanetary transfers. Comparison with alternative, up-to-date techniques points out that the new approach at hand is able to detect all the feasible transfer ellipses, in all cases, with very satisfactory accuracy in terms of final position error, even in challenging scenarios that include a huge number of revolutions or near-antipodal terminal positions. Full article

The uneven distribution of flow in water distribution networks (WDNs) can cause inefficient flows and pressure imbalances as well as degraded water quality in areas where demand is higher than the networks’ design limit. In this study, two faulty connections within a WDN in Greece that exhibited unusual geometric shapes favoring preferential flow paths were investigated. The three-dimensional computational fluid dynamics simulations were performed in ANSYS Fluent solver (v. 23.1) to study the internal behavior of this network for steady-state flows. The standard k-ε model was employed to calculate turbulence and energy losses in this network. In Connection A, which is a cross-shaped junction with two inlets and two outlets, and in Connection B, which is a complex 4 → 1 → 4 manifold connection, more than 80% of total inflow was found to be directed to a single outlet. The pressure contour plots revealed that this is due to the large total head losses associated with pronounced changes in flow direction. The role of explicit junction losses in network modeling and network improvements to improve hydraulic behavior has thereby gained prominence through this study. The applicability and capability of computational fluid dynamics in characterizing complex flow problems in urban WDNs have thereby proven to be significant. Full article

Circuit implementation is a widely accepted method for validating theoretical insights observed in chaotic systems. It also serves as a basis for numerous chaos-based engineering applications, including data encryption, random number generation, secure communication, neuromorphic computing, and so forth. To get feasible, compact, and cost-effective circuit implementations of chaotic systems, the underlying mathematical model may be simplified while preserving all rich nonlinear behaviors. In this framework, this manuscript presents a simplified Hopfield Neural Network (HNN) capable of generating a broad spectrum of complex behaviors using a minimal number of electronic elements. Based on Shil’nikov’s theorem for heteroclinic orbits, the number of non-zero synaptic connections in the matrix weights is reduced, while simultaneously using only one nonlinear activation function. As a result of these simplifications, we obtain the most compact electronic implementation of a tri-neuron HNN with the lowest component count but retaining complex dynamics. Comprehensive theoretical and numerical analyses by equilibrium points, density-colored continuation diagrams, basin of attraction, and Lyapunov exponents, confirm the presence of periodic oscillations, spiking, bursting, and chaos. Such chaotic dynamics range from single-scroll chaotic attractors to double-scroll chaotic attractors, as well as coexisting attractors to transient chaos. A brief security application of an S-Box utilizing the presented HNN is also given. Finally, a physical implementation of the HNN is given to confirm the proposed approach. Experimental observations are in good agreement with numerical results, demonstrating the usefulness of the proposed approach. Full article

The proposed work introduces a sizing algorithm to achieve a desired linear transconductance in the optimization of operational transconductance amplifiers (OTAs) by applying the $g_m/I_D$ method to find the initial width (W) and length (L) sizes of the transistors. These size values are used to run the non-dominated sorting genetic algorithm (NSGA-II) to perform a multi-objective optimization of three OTA topologies. The $g_m/I_D$ method begins with transistor characterization using MATLAB R2024a generated look-up tables (LUTs), which map normalized transconductance of the transistor channel dimensions, and key performance metrics of a complementary metal–oxide–semiconductor (CMOS) technology. The LUTs guide the initial population generation within NSGA-II during the optimization of OTAs to achieve not only a desired transconductance but also accuracy alongside linearity, high DC gain, low power consumption, and stability. The feasible W/L size solutions provided by NSGA-II are used to enhance the CMOS design of a memristor emulator, where the OTA with the desired transconductance is adapted to tune the behavior of the memristor, demonstrating improved pinched hysteresis loop characteristics. In addition, process, voltage and temperature (PVT) variations are performed by using TSMC 180 nm CMOS technology. The memristor-based on optimized OTAs is used to design a Leaky Integrate-and-Fire (LIF) neuron, which produces identical spike counts (seven spikes) under the same input conditions, though the time period varied with a CMOS inverter scaling. It is shown that increasing transistor widths by 100 in the inverter stage, the spike quantity is altered while changing the spiking period. This highlights the role of device sizing in modulating LIF neuron dynamics, and in addition, these findings provide valuable insights for energy-efficient neuromorphic hardware design. Full article

Diffusion on curved surfaces deviates from the flat case due to geometrical corrections in the evolution of its moments, such as the geodesic mean square displacement. Moreover, anomalous diffusion is widely used to model transport in disordered, confined, or crowded environments and can be described by a temporal subordination scheme, leading to a time-fractional diffusion equation. In this work, we analyze the dynamics of time subordinated anomalous diffusion on curved surfaces. By using a generalized Taylor expansion with fractional derivatives in the Caputo sense, we express the moments as a temporal power series and show that the anomalous exponent couples with curvature terms, leading to a competition between geometric and anomalous effects. This coupling indicates a mechanism through which curvature modulates anomalous transport. Full article

This research investigates the seismic behavior of continuous-span base-isolated bridges integrated with fluid inerter damper (FID) through a linear analytical framework under recorded earthquake excitations. The resisting mechanism of the FID is modelled as a combination of inertial and viscous forces, which are functions of the relative acceleration and velocity between connected nodes. Linear time-history simulations and a series of parametric analyses are conducted to examine how variations in inertance, damping ratio, and installation location affect key seismic response parameters, including deck acceleration, bearing displacement, and substructure base shear. Comparative analyses with conventional viscous dampers and isolation alone establish the relative effectiveness of FID. Analysis indicates that FID effectively reduces deck accelerations through apparent mass amplification, suppresses bearing displacements via viscous damping, and redistributes seismic forces depending on placement strategies. An optimum inertance range is identified that minimizes accelerations without amplifying base shear, with abutment-level placement proving most effective for pier shear control, while intermediate placement provides balanced reductions. Overall, FID consistently outperforms viscous dampers and conventional isolation, underscoring their potential as an advanced inerter-based solution for both new bridge design and retrofit applications. Full article

We provide an analytical framework for describing the propagation of light in waveguide arrays, considering both infinite and semi-infinite cases. The interaction up to second neighbors is taken into account, which provides a more realistic setup. We show that these solutions follow a distinctive structural pattern. This pattern reflects a transition from conventional Bessel functions to the lesser-known one-parameter generalized Bessel functions, offering new insights into the propagation dynamics in these systems. Full article

In this study, we explore the emergence of generalized synchronization (GS) in arrays of Hindmarsh–Rose (HR) neurons that are coupled through memristive synapses. We design coupling functions utilizing active memristors to facilitate GS in a bidirectionally coupled two-neuron memristive neural network (MNN). Our analysis employs a nearest neighbor (NN) approach. Our findings indicate that there is a threshold coupling strength for the active memristive synapses required to achieve GS. Additionally, we investigate how memristor parameters affect the temporal characteristics of synchronized neuronal firing patterns. Specifically, we discover that the interburst interval (IBI) is directly proportional to the coupling strength of the memristive synapses, while the interspike interval (ISI) is inversely proportional to this strength. Full article

In this paper, we study the different types of trajectories that correspond to a particular orbital behavior of caldera-type Hamiltonian systems. This particular orbital behavior is dynamical matching. Dynamical matching is an important chemical phenomenon that is encountered in many caldera-type organic chemical reactions. In this paper we will distinguish the different types of trajectories that correspond to this phenomenon using periodic orbit dividing surfaces. Full article