Dynamics

Ocean Thermal Energy Conversion (OTEC) is a process that can produce electricity by utilizing the temperature difference between deep cold water and surface warm water. The cold-water pipe (CWP) is a key component of OTEC systems, which transports deep cold water to the floating platform. The CWP is subjected to various environmental and operational loads, such as waves, currents, internal flow, and platform motion, which can affect its dynamic response and stability. In this paper, we establish a computational model of the mechanical performance of the CWP based on the Euler–Bernoulli beam theory and the Morrison equation, considering the effects of internal flow, sea current, and wave excitation. We use the differential quadrature method (DQM) to obtain a semi-analytical solution of the lateral displacement and bending moment of the CWP. We verify the correctness and validity of our model by comparing it with the finite element simulation results using OrcaFlex software. We also analyze the effects of operating conditions—such as wave intensity, clump weight at the bottom, and internal flow velocity—on the dynamic response of the CWP using numerical simulation and the orthogonal experimental method. The results show that changing the wave strength and internal flow velocity has little effect on the lateral displacement of the CWP but increasing the current velocity can significantly increase the lateral displacement of the CWP, which can lead to instability. The effects of waves, clump weight, internal flow, and sea current on the maximum bending moment of the CWP are similar; all of them increase sharply at first and then decrease gradually until they level off. The differences in the effects are mainly reflected in the different locations of the pipe sections. This paper suggests some design guidance for CWP in terms of dynamic responses depending on the operating conditions. This paper contributes to the journal’s scope by providing a novel and efficient method for analyzing the mechanical performance of CWP for OTEC systems, which is an important ocean energy resource. Full article

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions. Full article

To further explore the origins of Life, we consider three self-replicating chemical models. In general, models of the origin of Life include molecular components that can self-replicate and achieve exponential growth. Therefore, chemical self-replication is an essential chemical property of any model. The simplest self-replication mechanisms use the molecular product as a template for its synthesis. This mechanism is the so-called First-Order self-replication. Its regulatory limitations make it challenging to develop chemical networks, which are essential in the models of the origins of Life. In Second-Order self-replication, the molecular product forms a catalytic dimer capable of synthesis of the principal molecular product. In contrast with a simple template, the dimers show more flexibility in forming complex chemical networks since the chemical activity of the dimers can be activated or inhibited by the molecular components of the network. Here, we consider three minimal models: the First-Order Model (FOM), the Second-Order Model (SOM), and an Extended Second-Order Model (ESOM). We construct and analyze the mechanistic dimensionless ordinary differential equations (ODEs) associated with the models. The numerical integration of the set of ODEs gives us a visualization of these systems’ oscillatory behavior and compares their capacities for sustained autocatalytic behavior. The FOM model displays more complex oscillatory behavior than the ESOM model. Full article

The phenomena of electrical conductivity and electromigration in metallic systems are related, since in both cases the basic physical process is the scattering of conduction electrons by metal ions. Numerous searches have been made for equations connecting the conductivity with electromigration. In the case of a liquid metal, when using the Drude–Sommerfeld (DS) conductivity equation, it was not possible to obtain a quantitative relationship between these phenomena, which would be correct. Attempts to find such a relationship when taking into account the N. Mott correction (g-factor) in the DS equation were unsuccessful. This article proposes a different correction (b-factor) to the DS equation, which takes into account the possibility of varying the momentum transferred by the conduction electron to a metal ion during the scattering. This correction allows to establish a quantitative relationship between conductivity and electromigration as well as between electromigration in various binary systems with common components, in agreement with the experiment. The proposed theory describes well, in particular, two- and multi-component metal systems of any concentration (the consistency rule for triangles A–B, B–C, C–A). The value of the b-factor smoothly changes depending on the heat of vaporization of the metal, per unit volume. Full article

This study attempts to shed new light on the dynamics of a turning ship using the principles of free body diagrams (FBDs). Unexpectedly, the literature gap is defined by incomplete and flawed FBDs. The method behind this new approach involves the FBD of a turning ship, with all the essential forces included, namely propulsive force, sideward thruster force (producing the initial turning moment), drag force, lift force, centrifugal force, inertial force, and hydrodynamic force couple. From these forces, the force and moment equations are derived. The accelerations are calculated from the force and moment equilibria to simulate the dynamics from input parameters such as mass m, length L, draught D, and fluid density ρ. The turning dynamics are explained in terms of velocities, accelerations, forces, and moments, based on two conditions: flat and steep angles of attack (AoA) and long and short turning radii R. A critical result is the proportionality of lift and centrifugal forces, leading to the proposal of a pleometric index (m·L^–2·D^–1·ρ^–1), which is nonlinearly proportional to the product of AoA and R/L, characterising the dynamics of a turning ship. The FBD approach of this study also identified missing databases required for accurate simulation of turning dynamics, such as drag and lift coefficients of different hull geometries. Full article

Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath ($hb$), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution $W_{eq}$ generates orthogonal (Hermite) polynomials $H_n$ in momenta. Suitable moments $W_n$ of W (using the $H_n$’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the $W_n$’s yield irreversibly approach to equilibrium. The approach is extended (without $hb$) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ${\varphi }^4$ field theory (without $hb$). The extension to one non-relativistic quantum particle (with $hb$) employs the non-equilibrium Wigner function ($W_Q$): difficulties related to non-positivity of $W_Q$ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ${\varphi }^4$ field theory (a meson gas off-equilibrium, without $hb$), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ${\varphi }^4$ theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ${\varphi }^4$ one, yielding an approach to equilibrium. Full article

Since its original publication in 1978, Lozi’s chaotic map has been thoroughly explored and continues to be. Hundreds of publications have analyzed its particular structure or applied its properties in many fields (electronic devices such as memristors, A.I. with swarm intelligence, etc.). Several generalizations have been proposed, transforming the initial two-dimensional map into a multidimensional one. However, they do not respect the original constraint that allows this map to be one of the few strictly hyperbolic: a constant Jacobian. In this paper, we introduce a three-dimensional piece-wise linear extension respecting this constraint and we explore a special property never highlighted for chaotic mappings: the coexistence of thread chaotic attractors (i.e., attractors that are formed by a collection of lines) and sheet chaotic attractors (i.e., attractors that are formed by a collection of planes). This new three-dimensional mapping can generate a large variety of chaotic and hyperchaotic attractors. We give five examples of such behavior in this article. In the first three examples, there is the coexistence of thread and sheet chaotic attractors. However, their shapes are different and they are constituted by a different number of pieces. In the last two examples, the blow up of the attractors with respect to parameter a and b is highlighted. Full article

Quantum squeezing, an intriguing phenomenon that amplifies the uncertainty of one variable while diminishing that of its conjugate, may be studied as a time-dependent process, with exact solutions frequently derived from frameworks grounded in adiabatic invariants. Remarkably, we reveal that exact solutions can be ascertained in the presence of time-variant elastic forces, eschewing dependence on invariants or frozen eigenstate formalism. Delving into these solutions as an inverse problem unveils their direct connection to the design of elastic fields, responsible for inducing squeezing transformations onto canonical variables. Of particular note is that the dynamic transformations under investigation belong to a class of gentle quantum operations, distinguished by their delicate manipulation of particles, thereby circumventing the abrupt energy surges commonplace in conventional control protocols. Full article

Recent work has highlighted the vast array of dynamics possible within both neuronal networks and individual neural models. In this work, we demonstrate the capability of interacting chaotic Hindmarsh–Rose neurons to communicate and transition into periodic dynamics through specific interactions which we call mutual stabilization, despite individual units existing in chaotic parameter regimes. Mutual stabilization has been seen before in other chaotic systems but has yet to be reported in interacting neural models. The process of chaotic stabilization is similar to related previous work, where a control scheme which provides small perturbations on carefully chosen Poincaré surfaces that act as control planes stabilized a chaotic trajectory onto a cupolet. For mutual stabilization to occur, the symbolic dynamics of a cupolet are passed through an interaction function such that the output acts as a control on a second chaotic system. If chosen correctly, the second system stabilizes onto another cupolet. This process can send feedback to the first system, replacing the original control, so that in some cases the two systems are locked into persistent periodic behavior as long as the interaction continues. Here, we demonstrate how this process works in a two-cell network and then extend the results to four cells with potential generalizations to larger networks. We conclude that stabilization of different states may be linked to a type of information storage or memory. Full article

Ramsey theory constitutes the dynamics of mechanical systems, which may be described as abstract complete graphs. We address a mechanical system which is completely interconnected by two kinds of ideal Hookean springs. The suggested system mechanically corresponds to cyclic molecules, in which functional groups are interconnected by two kinds of chemical bonds, represented mechanically with two springs $k_1$ and $k_2$. In this paper, we consider a cyclic system (molecule) built of six equal masses m and two kinds of springs. We pose the following question: what is the minimal number of masses in such a system in which three masses are constrained to be connected cyclically with spring $k_1$ or three masses are constrained to be connected cyclically with spring $k_2$? The answer to this question is supplied by the Ramsey theory, formally stated as follows: what is the minimal number$R\left(3,3\right)?$ The result emerging from the Ramsey theory is $R\left(3,3\right)=6$. Thus, in the aforementioned interconnected mechanical system at least one triangle, built of masses and springs, must be present. This prediction constitutes the vibrational spectrum of the system. Thus, the Ramsey theory and symmetry considerations supply the selection rules for the vibrational spectra of the cyclic molecules. A symmetrical system built of six vibrating entities is addressed. The Ramsey approach works for 2D and 3D molecules, which may be described as abstract complete graphs. The extension of the proposed Ramsey approach to the systems, partially connected by ideal springs, viscoelastic systems and systems in which elasticity is of an entropic nature is discussed. “Multi-color systems” built of three kinds of ideal springs are addressed. The notion of the inverse Ramsey network is introduced and analyzed. Full article

Dual-rotating retarder polarimeters constitute a family of well-known instruments that are used today in a great variety of scientific and industrial contexts. In this work, the periodic intensity signal containing the information of all sixteen Mueller elements of depolarizing or nondepolarizing samples is determined for different ratios of angular velocities and non-ideal retarders, which are mathematically modeled with arbitrary retardances and take into account the possible diattenuating effect exhibited by both retarders. The alternative choices for generating a sufficient number of Fourier harmonics as well as their discriminating power are discussed. A general self-calibration procedure, which provides the effective values of the retardances and diattenuations of the retarders, the relative angles of the retarders and the analyzer, and the overall scale coefficient introduced by the detection and processing device are also described, leading to the absolute measurement of the Mueller matrix of the sample. Full article

This paper presents a nonlinear fault-tolerant vibration control system for a flexible arm, considering partial actuator fault. A lightweight flexible arm with lower stiffness will inevitably cause vibration which will impair the performance of the high-precision control system. Therefore, an operator-based robust nonlinear vibration control system is integrated by a double-sided interactive controller actuated by the Shape Memory Alloy (SMA) actuators for the flexible arm. Furthermore, to improve the safety and reliability of the safety-critical application, fault-tolerant dynamics for partial actuator fault are considered as an essential part of the proposed control system. The experimental cases are set to the partial actuator as faulty conditions, and the proposed vibration control scheme has fault-tolerant dynamics which can still effectively stabilize the vibration displacement. The reconfigurable controller improves the fault-tolerant performance by shortening the vibration time and reducing the vibration displacement of the flexible arm. In addition, compared with a PD controller, the proposed nonlinear vibration control has better performance than the traditional controller. The experimental results show that the effectiveness of the proposed method is confirmed. That is, the safety and reliability of the proposed fault-tolerant vibration control are verified even if in the presence of an actuator fault. Full article

The behavior of the network and its stability are governed by both dynamics of the individual nodes, as well as their topological interconnections. The attention mechanism as an integral part of neural network models was initially designed for natural language processing (NLP) and, so far, has shown excellent performance in combining the dynamics of individual nodes and the coupling strengths between them within a network. Despite the undoubted impact of the attention mechanism, it is not yet clear why some nodes of a network obtain higher attention weights. To come up with more explainable solutions, we tried to look at the problem from a stability perspective. Based on stability theory, negative connections in a network can create feedback loops or other complex structures by allowing information to flow in the opposite direction. These structures play a critical role in the dynamics of a complex system and can contribute to abnormal synchronization, amplification, or suppression. We hypothesized that those nodes that are involved in organizing such structures could push the entire network into instability modes and therefore need more attention during analysis. To test this hypothesis, the attention mechanism, along with spectral and topological stability analyses, was performed on a real-world numerical problem, i.e., a linear Multi-Input Multi-Output state-space model of a piezoelectric tube actuator. The findings of our study suggest that the attention should be directed toward the collective behavior of imbalanced structures and polarity-driven structural instabilities within the network. The results demonstrated that the nodes receiving more attention cause more instability in the system. Our study provides a proof of concept to understand why perturbing some nodes of a network may cause dramatic changes in the network dynamics. Full article

The damped van der Pol oscillator is a chaotic non-linear system. Small perturbations in initial conditions may result in wildly different trajectories. Controlling, or forcing, the behavior of a van der Pol oscillator is difficult to achieve through traditional adaptive control methods. Connecting two van der Pol oscillators together where the output of one oscillator, the driver, drives the behavior of its partner, the responder, is a proven technique for controlling the van der Pol oscillator. Deterministic artificial intelligence is a feedforward and feedback control method that leverages the known physics of the van der Pol system to learn optimal system parameters for the forcing function. We assessed the performance of deterministic artificial intelligence employing three different online parameter estimation algorithms. Our evaluation criteria include mean absolute error between the target trajectory and the response oscillator trajectory over time. Two algorithms performed better than the benchmark with necessary discussion of the conditions under which they perform best. Recursive least squares with exponential forgetting had the lowest mean absolute error overall, with a 2.46% reduction in error compared to the baseline, feedforward without deterministic artificial intelligence. While least mean squares with normalized gradient adaptation had worse initial error in the first 10% of the simulation, after that point it exhibited consistently lower error. Over the last 90% of the simulation, deterministic artificial intelligence with least mean squares with normalized gradient adaptation achieved a 48.7% reduction in mean absolute error compared to baseline. Full article

This work studies a two-time-scale functional system given by two jump diffusions under the scale separation by a small parameter $\epsilon \to 0$. The coefficients of the equations that govern the dynamics of the system depend on the segment process of the slow variable (responsible for capturing delay effects on the slow component) and on the state of the fast variable. We derive a moderate deviation principle for the slow component of the system in the small noise limit using the weak convergence approach. The rate function is written in terms of the averaged dynamics associated with the multi-scale system. The core of the proof of the moderate deviation principle is the establishment of an averaging principle for the auxiliary controlled processes associated with the slow variable in the framework of the weak convergence approach. The controlled version of the averaging principle for the jump multi-scale diffusion relies on a discretization method inspired by the classical Khasminkii’s averaging principle. Full article

The present paper is devoted to the solvability of various two-point boundary value problems for the equation $y^{\left(4\right)}=f(t,y,y^{\prime },y^{″},y^{‴}),$ where the nonlinearity f may be defined on a bounded set and is needed to be continuous on a suitable subset of its domain. The established existence results guarantee not just a solution to the considered boundary value problems but also guarantee the existence of monotone solutions with suitable signs and curvature. The obtained results rely on a basic existence theorem, which is a variant of a theorem due to A. Granas, R. Guenther and J. Lee. The a priori bounds necessary for the application of the basic theorem are provided by the barrier strip technique. The existence results are illustrated with examples. Full article

The damped oscillating structures recently revealed by a three parametric formula from the proton “effective” form factor data extracted of the measured total cross section ${\sigma }_{tot}^{bare}(e^{+}{e}^{-}\to p\overline{p})$ still seem to have an unknown origin. The conjectures of their direct manifestation of the quark-gluon structure of the proton indicate that they are not specific only of the proton and neutron, but they have to be one’s own, similar to other hadrons. Therefore, the oscillatory structures from the charged pion electromagnetic form factor timelike data, extracted of the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ are investigated by using the same procedure as in the case of the proton. The analysis shows the appearance of the oscillating structures in the description of the charged pion electromagnetic form factor timelike data by three parametric formula with a rather large value of ${\chi }^2/ndf$, while the description of the data by the physically well-founded Unitary and Analytic model has not revealed any damped oscillating structures. From the obtained result on the most simple object of strong interactions, one can conclude that damped oscillating structures received from the “effective” proton form factor data are probably generated by a utilization of the improper three parametric formula which does not describe these data with sufficient precision. Full article

In the cognitive and neural sciences, Bayesianism refers to a collection of concepts and methods stemming from various implementations of Bayes’ theorem, which is a formal way to calculate the conditional probability of a hypothesis being true based on prior expectations and updating priors in the face of errors. Bayes’ theorem has been fruitfully applied to describe and explain a wide range of cognitive and neural phenomena (e.g., visual perception and neural population activity) and is at the core of various theories (e.g., predictive processing). Despite these successes, we claim that Bayesianism has two interrelated shortcomings: its calculations and models are predominantly linear and noise is assumed to be random and unstructured versus deterministic. We outline ways that Bayesianism can address those shortcomings: first, by making more central the nonlinearities characteristic of biological cognitive systems, and second, by treating noise not as random and unstructured dynamics, but as the kind of structured nonlinearities of complex dynamical systems (e.g., chaos and fractals). We provide bistable visual percepts as an example of a real-world phenomenon that demonstrates the fruitfulness of integrating complex dynamical systems theory in Bayesian treatments of perception. Doing so facilitates a Bayesianism that is more capable of explaining a number of currently out-of-reach natural phenomena on their own, biologically realistic terms. Full article

The human brain is a complex network of connected neurons whose dynamics are difficult to describe. Brain dynamics are the global manifestation of individual neuron dynamics and the synaptic coupling between neurons. Membrane potential is a function of synaptic dynamics and electrophysiological coupling, with the parameters of postsynaptic potential, action potential, and ion pump dynamics. By modelling synaptic dynamics using physical laws and the time evolution of membrane potential using energy, neuron dynamics can be described. This local depiction can be scaled up to describe mesoscopic and macroscopic hierarchical complexity in the brain. Modelling results are favorably compared with physiological observation and physically acquired action potential profiles as reported in the literature. Full article