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24 pages, 1231 KiB

Open AccessArticle

Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

by Fei Luo, Yundong Li and Yi Xiang

Mathematics 2024, 12(5), 702; https://doi.org/10.3390/math12050702 - 28 Feb 2024

Viewed by 351

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and [...] Read more.

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at

(0, 0)

. Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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17 pages, 6961 KiB

Open AccessArticle

A New 3D Chaotic Attractor in Gene Regulatory Network

by Olga Kozlovska, Felix Sadyrbaev and Inna Samuilik

Mathematics 2024, 12(1), 100; https://doi.org/10.3390/math12010100 - 27 Dec 2023

Viewed by 659

Abstract

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D [...] Read more.

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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13 pages, 2691 KiB

Open AccessArticle

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

by Nikolay M. Evstigneev and Oleg I. Ryabkov

Mathematics 2023, 11(20), 4336; https://doi.org/10.3390/math11204336 - 18 Oct 2023

Viewed by 644

Abstract

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain [...] Read more.

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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11 pages, 1574 KiB

Open AccessArticle

The Third Type of Chaos in a System of Adaptively Coupled Phase Oscillators with Higher-Order Interactions

by Anastasiia A. Emelianova and Vladimir I. Nekorkin

Mathematics 2023, 11(19), 4024; https://doi.org/10.3390/math11194024 - 22 Sep 2023

Cited by 1 | Viewed by 634

Abstract

Adaptive network models arise when describing processes in a wide range of fields and are characterized by some specific effects. One of them is mixed dynamics, which is the third type of chaos in addition to the conservative and dissipative types. In this [...] Read more.

Adaptive network models arise when describing processes in a wide range of fields and are characterized by some specific effects. One of them is mixed dynamics, which is the third type of chaos in addition to the conservative and dissipative types. In this work, we consider a more complex type of connections between network elements—simplex, or higher-order adaptive interactions. Using numerical simulation methods, we analyze various characteristics of mixed dynamics and compare them with the case of pairwise couplings. We found that mixed dynamics in the case of simplex interactions is characterized by a very high similarity of a chaotic attractor to a chaotic repeller, as well as a stronger closeness of the sum of the Lyapunov exponents of the attractor and repeller to zero. This means that in the case of three elements, the conservative properties of the system are more pronounced than in the case of two. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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25 pages, 14145 KiB

Open AccessArticle

Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

by Nikolay M. Evstigneev and Nikolai A. Magnitskii

Mathematics 2023, 11(18), 3875; https://doi.org/10.3390/math11183875 - 11 Sep 2023

Cited by 1 | Viewed by 915

Abstract

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which [...] Read more.

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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13 pages, 1211 KiB

Open AccessArticle

Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse

by Yun Liu, Lifeng Guo and Xijuan Liu

Mathematics 2023, 11(15), 3321; https://doi.org/10.3390/math11153321 - 28 Jul 2023

Viewed by 550

Abstract

This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the [...] Read more.

This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the system and investigate the stability thresholds. Through the application of the center manifold theorem and bifurcation theory, our research has shown that the given model exhibits transcritical and flip bifurcation near its interior equilibrium point. The bifurcation diagrams, maximum Lyapunov exponents and phase portraits are presented to further substantiate the complexity. Finally, we present high-resolution stability diagrams that demonstrate the global structure of mode-locking oscillations. We also describe how these oscillations are interconnected and how their complexity unfolds as control parameters vary. The two parametric planes illustrate that the structure of Arnold’s tongues is based on the Stern–Brocot tree. This implies that the periodic occurrence of birth pulse and pulse fishing contributes to the development of more complex dynamical behaviors within the fish population. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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20 pages, 2939 KiB

Open AccessArticle

Universal Bifurcation Chaos Theory and Its New Applications

by Nikolai A. Magnitskii

Mathematics 2023, 11(11), 2536; https://doi.org/10.3390/math11112536 - 31 May 2023

Cited by 3 | Viewed by 1385

Abstract

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that [...] Read more.

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory. The obtained results once again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar–turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations and that chaos in the system can be confirmed only by detection of some main cycles or tori in accordance with the universal bifurcation diagram presented in the article. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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25 pages, 18293 KiB

Open AccessArticle

Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method

by Feiyun Pei, Guojiang Wu and Yong Guo

Mathematics 2023, 11(6), 1560; https://doi.org/10.3390/math11061560 - 22 Mar 2023

Cited by 1 | Viewed by 2103

Abstract

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of [...] Read more.

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics and Chaos: Theory and Application)

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