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9 pages, 267 KiB

Open AccessArticle

by Weizhen Quan, Tianxiu Lu, Risong Li, Yuanlin Chen, Xianfeng Ding and Yongjiang Li

Mathematics 2023, 11(23), 4763; https://doi.org/10.3390/math11234763 - 25 Nov 2023

Viewed by 863

Sensitive dependence on initial conditions is a crucial characteristic of chaos. The concept of measurable sensitivity (MS) was introduced as a measure-theoretic version of sensitive dependence on initial conditions. Their research demonstrated that MS arises from light mixing, indicates a finite number of [...] Read more.

Sensitive dependence on initial conditions is a crucial characteristic of chaos. The concept of measurable sensitivity (MS) was introduced as a measure-theoretic version of sensitive dependence on initial conditions. Their research demonstrated that MS arises from light mixing, indicates a finite number of eigenvalues for a transformation, and is not present in the case of infinite measure preservation. Unlike the traditional understanding of sensitivity, MS carries up to account for isomorphism in the sense of measure theory, which ignores the function’s behavior on null sets and eliminates dependence on the chosen metric. Inspired by the results of James on MS, this paper generalizes some of the concepts (including MS) that they used in their study of MS for conformal transformations to semi-flows, and generalizes their main results in this regard to semi-flows. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

12 pages, 784 KiB

Open AccessFeature PaperArticle

On the Double-Zero Bifurcation of Jerk Systems

by Cristian Lăzureanu

Mathematics 2023, 11(21), 4468; https://doi.org/10.3390/math11214468 - 28 Oct 2023

Cited by 2 | Viewed by 950

Abstract

In this paper, we construct approximate normal forms of the double-zero bifurcation for a two-parameter jerk system exhibiting a non-degenerate fold bifurcation. More precisely, using smooth invertible variable transformations and smooth invertible parameter changes, we obtain normal forms that are also jerk systems. [...] Read more.

In this paper, we construct approximate normal forms of the double-zero bifurcation for a two-parameter jerk system exhibiting a non-degenerate fold bifurcation. More precisely, using smooth invertible variable transformations and smooth invertible parameter changes, we obtain normal forms that are also jerk systems. In addition, we discuss some of their parametric portraits. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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15 pages, 1257 KiB

Open AccessArticle

On Hopf and Fold Bifurcations of Jerk Systems

by Cristian Lăzureanu and Jinyoung Cho

Mathematics 2023, 11(20), 4295; https://doi.org/10.3390/math11204295 - 15 Oct 2023

Cited by 7 | Viewed by 1214

Abstract

In this paper we consider a jerk system

\dot{x} = y, \dot{y} = z, \dot{z} = j (x, y, z, α),

where j is an arbitrary smooth function and

α

is a [...] Read more.

In this paper we consider a jerk system

\dot{x} = y, \dot{y} = z, \dot{z} = j (x, y, z, α),

where j is an arbitrary smooth function and

α

is a real parameter. Using the derivatives of j at an equilibrium point, we discuss the stability of that point, and we point out some local codim-1 bifurcations. Moreover, we deduce jerk approximate normal forms for the most common fold bifurcations. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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12 pages, 409 KiB

Open AccessArticle

Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop

by Erli Zhang and Stanford Shateyi

Mathematics 2023, 11(18), 3944; https://doi.org/10.3390/math11183944 - 17 Sep 2023

Viewed by 955

Abstract

This work revisits the number of limit cycles (LCs) in a piecewise smooth system of Hamiltonian with a heteroclinic loop generalization, subjected to perturbed functions through polynomials of degree m. By analyzing the asymptotic expansion (AE) of the Melnikov function with first-order [...] Read more.

This work revisits the number of limit cycles (LCs) in a piecewise smooth system of Hamiltonian with a heteroclinic loop generalization, subjected to perturbed functions through polynomials of degree m. By analyzing the asymptotic expansion (AE) of the Melnikov function with first-order

M (h)

near the generalized heteroclinic loop (HL), we utilize the expansions of the corresponding generators. This approach allows us to establish both lower and upper bounds for the quantity of limit cycles in the perturbed system. Our analysis involves a combination of expansion techniques, derivations, and divisions to derive these findings. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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22 pages, 774 KiB

Open AccessArticle

Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method

by Remus-Daniel Ene and Nicolina Pop

Mathematics 2023, 11(14), 3078; https://doi.org/10.3390/math11143078 - 12 Jul 2023

Cited by 4 | Viewed by 964

Abstract

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, [...] Read more.

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, the closed-form solutions can be established. The obtained solutions have a periodical behavior. These geometrical properties allow reducing the initial system to a second-order nonlinear differential equation. The latter equation is solved analytically using the OHPM procedure. The validation of the OHPM method is presented for three cases of the physical parameters. The advantages of the OHPM technique, such as the small number of iterations (the efficiency), the convergence control (in the sense that the semi-analytical solutions are approaching the exact solution), and the writing of the solutions in an effective form, are shown graphically and with tables. The accuracy of the results provides good agreement between the analytical and corresponding numerical results. Other dynamic systems with similar geometrical properties could be successfully solved using the same procedure. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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13 pages, 461 KiB

Open AccessArticle

Stabilization and Chaos Control of an Economic Model via a Time-Delayed Feedback Scheme

by Yang Hu and Guangping Hu

Mathematics 2023, 11(13), 2994; https://doi.org/10.3390/math11132994 - 5 Jul 2023

Cited by 2 | Viewed by 1089

Abstract

This paper addresses the problem of chaos control in an economic mathematical dynamical model. By regarding the control variables as the bifurcation parameters, the stability of equilibria and the existence of Hopf bifurcations of the relevance feedback system are investigated, and the criterion [...] Read more.

This paper addresses the problem of chaos control in an economic mathematical dynamical model. By regarding the control variables as the bifurcation parameters, the stability of equilibria and the existence of Hopf bifurcations of the relevance feedback system are investigated, and the criterion of controllability for the chaotic system is obtained based on a time-delayed feedback control technique. Furthermore, numerical simulations are provided to demonstrate the feasibility of our methods and results. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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19 pages, 670 KiB

Open AccessArticle

High Order Energy Preserving Composition Method for Multi-Symplectic Sine-Gordon Equation

by Jianqiang Sun, Jingxian Zhang and Jiameng Kong

Mathematics 2023, 11(5), 1105; https://doi.org/10.3390/math11051105 - 22 Feb 2023

Cited by 1 | Viewed by 1208

Abstract

A fourth-order energy preserving composition scheme for multi-symplectic structure partial differential equations have been proposed. The accuracy and energy conservation properties of the new scheme were verified. The new scheme is applied to solve the multi-symplectic sine-Gordon equation with periodic boundary conditions and [...] Read more.

A fourth-order energy preserving composition scheme for multi-symplectic structure partial differential equations have been proposed. The accuracy and energy conservation properties of the new scheme were verified. The new scheme is applied to solve the multi-symplectic sine-Gordon equation with periodic boundary conditions and compared with the corresponding second-order average vector field scheme and the second-order Preissmann scheme. The numerical experiments show that the new scheme has fourth-order accuracy and can preserve the energy conservation properties well. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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13 pages, 472 KiB

Open AccessArticle

Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method

by Remus-Daniel Ene, Nicolina Pop, Marioara Lapadat and Luisa Dungan

Mathematics 2022, 10(21), 4118; https://doi.org/10.3390/math10214118 - 4 Nov 2022

Cited by 4 | Viewed by 1553

Abstract

This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions [...] Read more.

This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the optimal homotopy asymptotic method (OHAM). These represent the

ε

-approximate OHAM solutions. A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure is suitable to be applied for dynamical systems with certain geometrical properties. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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17 pages, 2016 KiB

Open AccessArticle

Dynamics of a Reduced System Connected to the Investigation of an Infinite Network of Identical Theta Neurons

by Lavinia Bîrdac, Eva Kaslik and Raluca Mureşan

Mathematics 2022, 10(18), 3245; https://doi.org/10.3390/math10183245 - 7 Sep 2022

Cited by 1 | Viewed by 1207

Abstract

We consider an infinite network of identical theta neurons, all-to-all coupled by instantaneous synapses. Using the Watanabe–Strogatz Ansatz, the mathematical model of this infinite network is reduced to a two-dimensional system of differential equations. We determine the number of equilibria of this reduced [...] Read more.

We consider an infinite network of identical theta neurons, all-to-all coupled by instantaneous synapses. Using the Watanabe–Strogatz Ansatz, the mathematical model of this infinite network is reduced to a two-dimensional system of differential equations. We determine the number of equilibria of this reduced system with respect to two characteristic parameters. Furthermore, we discuss the stability properties of each equilibrium and the possible bifurcations that may take place. As a result, the occurrence of exotic higher codimension bifurcations involving a degenerate center is also unveiled. Numerical results are also presented to illustrate complex dynamic behaviour in the reduced system. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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28 pages, 17111 KiB

Open AccessArticle

New Insights into a Three-Sub-Step Composite Method and Its Performance on Multibody Systems

by Yi Ji, Huan Zhang and Yufeng Xing

Mathematics 2022, 10(14), 2375; https://doi.org/10.3390/math10142375 - 6 Jul 2022

Cited by 6 | Viewed by 1788

Abstract

This paper develops a new implicit solution procedure for multibody systems based on a three-sub-step composite method, named TTBIF (trapezoidal–trapezoidal backward interpolation formula). The TTBIF is second-order accurate, and the effective stiffness matrices of the first two sub-steps are the same. In this [...] Read more.

This paper develops a new implicit solution procedure for multibody systems based on a three-sub-step composite method, named TTBIF (trapezoidal–trapezoidal backward interpolation formula). The TTBIF is second-order accurate, and the effective stiffness matrices of the first two sub-steps are the same. In this work, the algorithmic parameters of the TTBIF are further optimized to minimize its local truncation error. Theoretical analysis shows that for both undamped and damped systems, this optimized TTBIF is unconditionally stable, controllably dissipative, third-order accurate, and has no overshoots. Additionally, the effective stiffness matrices of all three sub-steps are the same, leading to the effective stiffness matrix being factorized only once in a step for linear systems. Then, the implementation procedure of the present optimized TTBIF for multibody systems is presented, in which the position constraint equation is strictly satisfied. The advantages in accuracy, stability, and energy conservation of the optimized TTBIF are validated by some benchmark multibody dynamic problems. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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14 pages, 12935 KiB

Open AccessArticle

Asymmetric Growth of Tumor Spheroids in a Symmetric Environment

by Meitham Amereh, Yakine Bahri, Roderick Edwards, Mohsen Akbari and Ben Nadler

Mathematics 2022, 10(12), 1955; https://doi.org/10.3390/math10121955 - 7 Jun 2022

Cited by 4 | Viewed by 1540

Abstract

In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The [...] Read more.

In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The analytical solution of the governing model was presented in an orthonormal spherical harmonic basis. It was shown that the radially symmetric steady-state solution to the growth of tumor spheroids, under symmetric growth conditions, was unstable with respect to small asymmetric perturbations. Such perturbations excited the asymmetric modes of growth, which could grow in time and change the spherical configuration of the tumor. The number of such modes and their rates of growth depended on parameters such as surface tension, external energy and the rate of nutrient consumption. This analysis indicated that the spherical configuration of tumor spheroids, even under experimentally controlled symmetric growth conditions, were naturally unstable. This was confirmed by a comparison between the shapes of in vitro human glioblastoma (hGB) spheroids and the configuration of the first few asymmetric modes predicted by the model. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

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