Dynamics, Volume 3, Issue 2 (June 2023) – 8 articles

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