Symmetry in Nonlinear Dynamics and Chaos II

Nonlinear dynamics and chaos are vibrant fields of physics that have enhanced our understanding of various phenomena. This Special Issue reviews recent advancements in nonlinear dynamical systems and chaotic behavior. It includes 22 papers recently published in MDPI’s Symmetry under the thematic title “Symmetry in Nonlinear Dynamics and Chaos II”. One of these papers is a review article [1], which provides a comprehensive overview of different types of intermittency. Additionally, it explains two recent formulations for evaluating reinjection processes. These new theoretical formulations have explained several previously termed pathological tests. The theoretical background also addresses the effects of noise on the reinjection mechanism [2].

In Ref. [3], Bogoi and coauthors examine a system of second-order stochastic ordinary differential equations (SDEs) related to satellite motion. This paper evaluates the accuracy and stability of five finite difference schemes for solving these SDEs. The authors discuss the uniformity of stabilization behavior in the stochastic trajectories of a dynamical system and analyze the impact of noise on the results. The graphical representations of the SDEs presented in this paper illustrate the symmetry of stochastic trajectories around the solution of the deterministic system.

Junjie Tang and coauthors analyze the decline in lily production caused by weed infestation [4]. They propose an advanced weed detection method that combines symmetry-based convolutional neural networks [5] with metaheuristic optimization. To improve detection efficiency, they introduce an optimized YOLOv7-Tiny model, which incorporates dynamic pruning and knowledge distillation. This approach reduces computational complexity while maintaining high accuracy. The study presents a high-performance, lightweight, and scalable solution for real-time precision weed management in lily fields, offering valuable insights for agricultural automation and smart farming applications.

In Ref. [6], Prykarpatski and coauthors investigate the superization problem in integrable nonlinear dynamical systems on functional manifolds. They focus on a quantum many-particle Schrödinger–Davydov model defined on an axis. The authors demonstrate that a broad class of classical Lax-type integrable nonlinear dynamical systems on axes exists, for which a superization scheme can be effectively developed. This scheme involves a reasonable superization of the related Lax-type representation by transitioning from the basic algebra of pseudo-differential operators on the axis to the corresponding super-algebra of super-pseudo-differential operators on the super-axis.

Following previous studies [7], Shams and Carpentieri present a new fractional parallel technique for solving nonlinear equations [8]. By employing a dynamical systems approach, they identify optimal parameter values. The symmetry observed in the dynamical planes for various fractional parameters demonstrates the stability of the method and its effectiveness in addressing nonlinear problems. This approach has been successfully applied to several engineering challenges.

Sarıkaya and coauthors present a novel metaheuristic optimization method that combines the Henry Gas Solubility Optimization (HGSO) technique with symmetric chaotic systems [9]. This new approach is called Chaotic Henry Gas Solubility Optimization (CHGSO), and its primary objective is to enhance the performance of the HGSO method. The authors applied CHGSO to 47 benchmark functions to optimize parameters for a PID controller used in the speed control of a DC motor. The results demonstrate that the proposed method achieved the best performance in 43 of the benchmark functions, surpassing other algorithms.

In Ref. [10], Dragan and Popa discuss the linear quadratic (LQ) optimal control problem for stochastic systems that are controlled by impulses over an infinite time horizon. They demonstrate that the well-posedness of the optimal control problem is assured by the existence of a maximal and bounded solution to the associated backward jump linear matrix difference equation (BJMLDE) that incorporates a Riccati-type jumping operator. Furthermore, the authors indicate that if the BJMLDE with a Riccati-type jumping operator has a maximal solution that meets a specific sign condition, then the optimal control problem is feasible. This feasibility is contingent on either having an optimal control in a state feedback form or the maximal solution of the BJMLDE being a stabilizing solution.

Almatroud and collaborators introduce a new third-order symmetric difference equation, which has been transformed into a 3D discrete symmetric map [11]. They demonstrate that this map exhibits several nonlinear characteristics, including multistability, chaos, and hyperchaos [12]. Additionally, they introduce a nonlinear controller designed to stabilize the symmetric map and synchronize two unified symmetric maps.

In Ref. [13], Yang and coauthors investigate the synchronization issue between complex dynamic networks [14] and chaotic systems [15]. They propose a state estimation scheme for multi-layer complex networks by using a state observer. The authors analyze a specific class of complex dynamic networks where the nodes correspond one-to-one. They assess the stability of the system by using the primary stability function analysis method and demonstrate the effectiveness of the network state observer in achieving synchronization between the target system and the complex dynamic network.

Navascués and Mohapatra introduce new forms of contractivity that extend the traditional concept of Banach contraction [16]. They also provide sufficient conditions for the existence of fixed points and common fixed points. The authors expand, to some degree, the modern theory of fixed points and common fixed points for self-maps defined on b-metric and quasi-normed spaces, along with their approximation methods.

The determination of the vulnerability of concrete buildings at slopes under the action of earthquakes is a topic of current research [17]. Vielma-Quintero and coauthors investigate buildings situated on slopes in densely populated urban areas, a common situation in Latin American cities that face high seismic risks [18]. They examine the asymmetry of medium-height reinforced concrete-frame buildings on variable inclines and how this affects their nonlinear response, evaluated through displacements, rotations, and damage. The study reveals the significant influence of structural asymmetry on nonlinear response parameters, such as ductility, transient inter-story drifts, and roof rotations. Additionally, it uncovers demand distributions for structural elements that exceed traditional analysis and expectations in earthquake-resistant design.

Coupled processes of deformation in media, considering heat and mass transfer, have been subjects of scientific interest in recent years [19,20]. Lurie and coauthors have developed a variational model for irreversible deformation processes that encompasses a broader range of coupled effects [21]. Their approach introduces non-integrable variational forms that clearly distinguish between dissipative processes and reversible deformation processes. They have established the fundamental nature of the symmetry and anti-symmetry properties of tensors related to multi-indices, which characterize independent arguments in the bi-linear forms used within the variational formulation of thermo-mechanical models. Additionally, an algorithm for creating variational models of dissipative irreversible processes has been proposed. This algorithm involves identifying the necessary number of dissipative channels and incorporating them into the established model of a reversible process.

In Ref. [22], Moysis and collaborators investigate the task of constructing maps, where statistical measures, such as the mean value, can be effectively controlled by adjusting the parameters of the map. They consider a family of piecewise maps, which include three control parameters that influence the endpoint interpolations. These maps can achieve a diverse range of values for their statistical mean. To illustrate this, the authors present chaotic path surveillance as a potential application for the maps they have designed. As an application case, the authors propose a strategy for the chaotic maneuvering of an unmanned aerial vehicle [23].

Bogoi and coauthors investigate the weak and strong convergence of six numerical schemes applied to multiplicative noise, additive noise, and a system of stochastic differential equations [24]. Among these methods, the Efficient Runge–Kutta technique [25] stands out as the top performer, demonstrating the best convergence properties across all scenarios, including the challenging case of multiplicative noise. The authors consider that this finding highlights the importance of developing advanced numerical techniques specifically tailored for stochastic systems.

Verma, Sumelka, and Yadav propose an approximation algorithm that utilizes Legendre and Chebyshev artificial neural networks to explore approximate solutions for fractional Lienard and Duffing equations involving a Caputo fractional derivative [26]. Their approach transforms the given nonlinear fractional differential equation into an unconstrained minimization problem. They also examine several non-integer order problems to verify their theoretical results. The numerical results demonstrate that the proposed strategies achieve superior outcomes compared to existing numerical techniques [27,28,29].

In Ref. [30], Matouk and collaborators explore the complex dynamics of a fractional-order multi-scroll chaotic system based on the extended Gamma function. They begin by introducing the extended left and right Caputo fractional differential operators. The authors propose a new operator that incorporates an additional fractional parameter, providing a higher degree of freedom. This enhancement allows the system to exhibit a greater variety of complex dynamics. The rich and complex dynamics observed include coexisting one-scroll chaotic attractors, two-scroll chaotic attractors, and approximate periodic cycles. These dynamics are shown to persist for a shorter duration compared to the corresponding states of the integer-order version of the multi-scroll system.

Potirakis and coauthors confirm that a recently introduced symbolic time-series-analysis method, known as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately obtain the exponent of the distribution of waiting times in the symbolic dynamics of two symbols generated by the 3D Ising model when it is in a critical state [31]. Following the numerical validation of PNA-STSM’s reliability, three examples are presented to demonstrate its application to systems that exhibit on–off intermittency dynamics [32]. The findings suggest that identifying a system’s proximity to spontaneous symmetry breaking offers a deeper understanding of its dynamics in terms of criticality.

Khan and collaborators introduce a nonlinear dynamical system for leukemia that utilizes a piecewise modified ABC fractional-order derivative [33]. They analyze this system through both theoretical and computational methods, focusing on the crossover effects of the model. The numerical analysis relies on a computational method developed by the authors, which has been implemented using Lagrange’s interpolation polynomial. The newly constructed model is studied to determine the existence of solutions, its stability, and to present computational results. The authors suggest expanding their research by integrating the stochastic version, using Ref. [34] as a guiding resource.

In Ref. [35], Lawnik and coauthors propose a family of piecewise chaotic maps. This family is characterized by the nonlinear functions used for each piece of the map, which can be either symmetric or asymmetric. By imposing constraints on the shape of each piece, the resulting maps do not have equilibria and can exhibit chaotic behavior. As such, this family falls under the category of systems with hidden attractors [36]. The authors provide several examples of chaotic maps that demonstrate fractal-like, symmetrical patterns at the transition between chaotic and non-chaotic behavior. The authors are working to extend the methodology to a broader range of multi-dimensional maps. The design constraint regarding the absence of fixed points must be met by every state of the map.

Moysis and collaborators introduce a chaotification technique designed to increase the complexity of chaotic maps [37]. This technique is an extension of those developed in Ref. [38]. It involves combining the results from multiple scaling of the map’s value for the next iteration. The maps produced using this approach can attain a high Lyapunov exponent value, which can be adjusted through careful parameter tuning. As a potential application for these transformed maps, the encryption of B-spline curves and patches are explored.

Agop and coauthors provide a description of complex systems using a holographic perspective [39]. To achieve this, they utilize Scale Relativity Theory, applying Schrödinger- and Madelung-type scenarios to the dynamics of these systems. This approach reveals a Riccati-type gauge invariance that plays a crucial role in the dynamics of complex systems. As a result, it gives rise to conservation laws, simultaneity, and synchronization among the structural units’ dynamics, as well as temporal patterns through harmonic mappings.

Elaskar and collaborators introduce a methodology to study the reinjection process in type V intermittency [40]. The authors implement the Perron–Frobenius operator technique [41]. Using this approach, they calculate the reinjection probability density function (RPD) and the probability density of the laminar lengths. They analyze a family of maps with both discontinuous and continuous RPDs. Several tests are conducted to compare the proposed technique with the classical theory of intermittency, the M function methodology, and numerical data. The new technique can accurately capturenumerical data. Thus, the scheme presented is a useful tool for theoretically evaluating the statistical variables related to type V intermittency.