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Mathematics
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Nonlinear Dynamics, Chaos and Complex Systems
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Nonlinear Dynamics, Chaos and Complex Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 4583

Special Issue Editor

Prof. Dr. José Roberto Castilho Piqueira

E-Mail Website
Guest Editor
Polytechnic School, University of São Paulo, São Paulo 05508-010, SP, Brazil
Interests: bifurcation theory; center manifolds

Special Issue Information

Dear Colleagues,

Talking about dynamics is often challenging because it has gained importance in understanding natural phenomena through differential equations proposed by Newton and Leibniz at the start of the 17th century.

Following these seminal works, several fields of natural sciences developed models and ways to describe their study objects by considering how the various observable variables evolve depending on temporal and spatial dynamics.

Starting with ordinary equations for mechanical problems, the idea was to provide exact analytical solutions that, until now, are difficult to obtain in a general fashion.

Considering the broader development of engineering and physics during the 18th and 19th centuries, continuous fluid mechanics and electromagnetism were characterized by partial derivative equations with rigorous mathematical formulations, proposed by Navier, Stokes, and Maxwell.

At the start of the 20th century, Poincaré formulated the three-bodies problem, wherein initial proximity of the bodies could lead to divergence due to nonlinearities.

Connecting this idea with the turbulence phenomenon, Ruelle and Takens proposed that nonlinearities and high state space dimensions would be responsible for this intriguing nature manifestation.

Nowadays, chaos, initial condition sensitivity, noise due to nonlinearities, unpredictability, and complexity appear in all science studies: meteorology, physics, engineering, chemistry, biology, and social sciences; nonlinear dynamics is ubiquitous.

This Special Issue welcomes the inclusion of various approaches and study areas, which have been thoroughly examined and discussed.

Prof. Dr. José Roberto Castilho Piqueira
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • bifurcations
  • center manifolds
  • chaotic behaviors
  • dynamical models
  • nonlinear oscillations

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Published Papers (4 papers)

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Research

18 pages, 8297 KiB  
Article
Adaptive Asymptotic Shape Synchronization of a Chaotic System with Applications for Image Encryption
by Yangxin Luo, Yuanyuan Huang, Fei Yu, Diqing Liang and Hairong Lin
Mathematics 2025, 13(1), 128; https://doi.org/10.3390/math13010128 - 31 Dec 2024
Cited by 1 | Viewed by 553
Abstract
In contrast to previous research that has primarily focused on distance synchronization of states in chaotic systems, shape synchronization emphasizes the geometric shape of the attractors of two chaotic systems. Diverging from the existing work on shape synchronization, this paper introduces the application [...] Read more.
In contrast to previous research that has primarily focused on distance synchronization of states in chaotic systems, shape synchronization emphasizes the geometric shape of the attractors of two chaotic systems. Diverging from the existing work on shape synchronization, this paper introduces the application of adaptive control methods to achieve asymptotic shape synchronization for the first time. By designing an adaptive controller using the proposed adaptive rule, the response system under control is able to attain asymptotic synchronization with the drive system. This method is capable of achieving synchronization for models with parameters requiring estimation in both the drive and response systems. The control approach remains effective even in the presence of uncertainties in model parameters. The paper presents relevant theorems and proofs, and simulation results demonstrate the effectiveness of adaptive asymptotic shape synchronization. Due to the pseudo-random nature of chaotic systems and their extreme sensitivity to initial conditions, which make them suitable for information encryption, a novel channel-integrated image encryption scheme is proposed. This scheme leverages the shape synchronization method to generate pseudo-random sequences, which are then used for shuffling, scrambling, and diffusion processes. Simulation experiments demonstrate that the proposed encryption algorithm achieves exceptional performance in terms of correlation metrics and entropy, with a competitive value of 7.9971. Robustness is further validated through key space analysis, yielding a value of 10210×2512, as well as visual tests, including center and edge cropping. The results confirm the effectiveness of adaptive asymptotic shape synchronization in the context of image encryption. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos and Complex Systems)
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14 pages, 542 KiB  
Article
Hidden-like Attractors in a Class of Discontinuous Dynamical Systems
by Hany A. Hosham, Mashael A. Aljohani, Eman D. Abou Elela, Nada A. Almuallem and Thoraya N. Alharthi
Mathematics 2024, 12(23), 3784; https://doi.org/10.3390/math12233784 - 29 Nov 2024
Viewed by 790
Abstract
In continuous dynamical systems, a hidden attractor occurs when its basin of attraction does not connect with small neighborhoods of equilibria. This research aims to investigate the presence of hidden-like attractors in a class of discontinuous systems that lack equilibria. The nature of [...] Read more.
In continuous dynamical systems, a hidden attractor occurs when its basin of attraction does not connect with small neighborhoods of equilibria. This research aims to investigate the presence of hidden-like attractors in a class of discontinuous systems that lack equilibria. The nature of non-smoothness in Filippov systems is critical for producing a wide variety of interesting dynamical behaviors and abrupt transient responses to dynamic processes. To show the effects of non-smoothness on dynamic behaviors, we provide a simple discontinuous system made of linear subsystems with no equilibria. The explicit closed-form solutions for each subsystem have been derived, and the generalized Poincaré maps have been established. Our results show that the periodic orbit can be completely established within a sliding region. We then carry out a mathematical investigation of hidden-like attractors that exhibit sliding-mode characteristics, particularly those associated with grazing-sliding behaviors. The proposed system evolves by adding a nonlinear function to one of the vector fields while still preserving the condition that equilibrium points do not exist in the whole system. The results of the linear system are useful for investigating the hidden-like attractors of flow behavior across a sliding surface in a nonlinear system using numerical simulation. The discontinuous behaviors are depicted as motion in a phase space governed by various hidden attractors, such as period doubling, period-m segments, and chaotic behavior, with varying interactions with the sliding mode. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos and Complex Systems)
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21 pages, 11071 KiB  
Article
Dynamical Analysis and Sliding Mode Controller for the New 4D Chaotic Supply Chain Model Based on the Product Received by the Customer
by Muhamad Deni Johansyah, Sundarapandian Vaidyanathan, Aceng Sambas, Khaled Benkouider, Seyed Mohammad Hamidzadeh and Monika Hidayanti
Mathematics 2024, 12(13), 1938; https://doi.org/10.3390/math12131938 - 22 Jun 2024
Cited by 3 | Viewed by 1016
Abstract
Supply chains comprise various interconnected components like suppliers, manufacturers, distributors, retailers, and customers, each with unique variables and interactions. Managing dynamic supply chains is highly challenging, particularly when considering various sources of risk factors. This paper extensively explores dynamical analysis and multistability analysis [...] Read more.
Supply chains comprise various interconnected components like suppliers, manufacturers, distributors, retailers, and customers, each with unique variables and interactions. Managing dynamic supply chains is highly challenging, particularly when considering various sources of risk factors. This paper extensively explores dynamical analysis and multistability analysis to understand nonlinear behaviors and pinpoint potential risks within supply chains. Different phase portraits are used to demonstrate the impact of various factors such as transportation risk, quality risk, distortion, contingency reserves, and safety stock on both customers and retailers. We introduced a sliding mode control method that computes the sliding surface and its derivative by considering the error and its derivative. The equivalent control law based on the sliding surface and its derivative is derived and validated for control purposes. Our results show that the controller SMC can significantly enhance supply chain stability and efficiency. This research provides a robust framework for understanding complex supply chain dynamics and offers practical solutions to enhance supply chain resilience and flexibility. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos and Complex Systems)
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13 pages, 1313 KiB  
Article
A Recurrent Neural Network for Identifying Multiple Chaotic Systems
by José Luis Echenausía-Monroy, Jonatan Pena Ramirez, Joaquín Álvarez, Raúl Rivera-Rodríguez, Luis Javier Ontañón-García and Daniel Alejandro Magallón-García
Mathematics 2024, 12(12), 1835; https://doi.org/10.3390/math12121835 - 13 Jun 2024
Cited by 3 | Viewed by 1569
Abstract
This paper presents a First-Order Recurrent Neural Network activated by a wavelet function, in particular a Morlet wavelet, with a fixed set of parameters and capable of identifying multiple chaotic systems. By maintaining a fixed structure for the neural network and using the [...] Read more.
This paper presents a First-Order Recurrent Neural Network activated by a wavelet function, in particular a Morlet wavelet, with a fixed set of parameters and capable of identifying multiple chaotic systems. By maintaining a fixed structure for the neural network and using the same activation function, the network can successfully identify the three state variables of several different chaotic systems, including the Chua, PWL-Rössler, Anishchenko–Astakhov, Álvarez-Curiel, Aizawa, and Rucklidge models. The performance of this approach was validated by numerical simulations in which the accuracy of the state estimation was evaluated using the Mean Square Error (MSE) and the coefficient of determination (r2), which indicates how well the neural network identifies the behavior of the individual oscillators. In contrast to the methods found in the literature, where a neural network is optimized to identify a single system and its application to another model requires recalibration of the neural algorithm parameters, the proposed model uses a fixed set of parameters to efficiently identify seven chaotic systems. These results build on previously published work by the authors and advance the development of robust and generic neural network structures for the identification of multiple chaotic oscillators. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos and Complex Systems)
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