Fractal and Fractional

Research

13 pages, 294 KiB

Open AccessArticle

Positive Solutions for Dirichlet BVP of PDE Involving \({\varphi_{p}}\)-Laplacian

by Feng Xiong and Wentao Huang

Fractal Fract. 2024, 8(3), 130; https://doi.org/10.3390/fractalfract8030130 - 23 Feb 2024

Viewed by 1131

Abstract

In this paper, we investigate the existence of infinitely many small solutions for problem

(f_{φ_{p}})

involving

φ_{p}

-Laplacian by exploiting critical point theory. Moreover, the present study first attempts to address discrete Dirichlet problems with

φ_{p}

-Laplacian [...] Read more.

In this paper, we investigate the existence of infinitely many small solutions for problem

(f_{φ_{p}})

involving

φ_{p}

-Laplacian by exploiting critical point theory. Moreover, the present study first attempts to address discrete Dirichlet problems with

φ_{p}

-Laplacian in relation to some relative existing references. As far as we know, this research of the partial discrete bvp involves

φ_{p}

-Laplacian for the first time. Our results are illustrated with three examples. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

18 pages, 337 KiB

Open AccessArticle

Decay Properties for Transmission System with Infinite Memory and Distributed Delay

by Hicham Saber, Abdelkader Braik, Noureddine Bahri, Abderrahmane Beniani, Tariq Alraqad, Yousef Jawarneh and Khaled Zennir

Fractal Fract. 2024, 8(2), 94; https://doi.org/10.3390/fractalfract8020094 - 31 Jan 2024

Viewed by 1225

Abstract

We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under [...] Read more.

We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under suitable assumption on the geometrical conditions on the damping, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

16 pages, 300 KiB

Open AccessArticle

Existence Result for Coupled Random First-Order Impulsive Differential Equations with Infinite Delay

by Abdelkader Moumen, Fatima Zohra Ladrani, Mohamed Ferhat, Amin Benaissa Cherif, Mohamed Bouye and Keltoum Bouhali

Fractal Fract. 2024, 8(1), 10; https://doi.org/10.3390/fractalfract8010010 - 21 Dec 2023

Viewed by 1133

Abstract

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under [...] Read more.

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on

Y_{1}

,

Y_{2}

and

ϖ_{1}

,

ϖ_{2}

, which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

23 pages, 7925 KiB

Open AccessArticle

Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

by Rong Liu and Huilin Shang

Fractal Fract. 2023, 7(12), 888; https://doi.org/10.3390/fractalfract7120888 - 18 Dec 2023

Cited by 1 | Viewed by 1331

Abstract

This study’s objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, [...] Read more.

This study’s objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, static bifurcations of its nondimensional system are discussed. It is found that variations of two structural parameters can induce different numbers and natures of potential wells. Next, the cases of mono-potential wells and double wells are explored. The forms and stabilities of the resonant responses within each potential well and the inter-well resonant responses are discussed via different theoretical methods. The results show that the natural frequencies and trends of frequency responses in the cases of mono- and double-potential wells are totally different; as a result of the saddle-node bifurcations of resonant solutions, raising the excitation level or frequency can lead to the coexistence of bistable responses within each well and cause an inter-well periodic response. Moreover, in addition to verifying the accuracy of the theoretical prediction, numerical results considering the disturbance of initial conditions are presented to detect complicated dynamical behaviors such as jump between coexisting resonant responses, intra-well period-two responses and chaos. The results herein provide a theoretical foundation for designing and utilizing the multi-stable behaviors of irrationally nonlinear oscillators. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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13 pages, 1245 KiB

Open AccessArticle

The Analytical Stochastic Solutions for the Stochastic Potential Yu–Toda–Sasa–Fukuyama Equation with Conformable Derivative Using Different Methods

by Sahar Albosaily, Elsayed M. Elsayed, M. Daher Albalwi, Meshari Alesemi and Wael W. Mohammed

Fractal Fract. 2023, 7(11), 787; https://doi.org/10.3390/fractalfract7110787 - 28 Oct 2023

Cited by 10 | Viewed by 1147

Abstract

We consider in this study the (3+1)-dimensional stochastic potential Yu–Toda–Sasa–Fukuyama with conformable derivative (SPYTSFE-CD) forced by white noise. For different kind of solutions of SPYTSFE-CD, including hyperbolic, rational, trigonometric and function, we use He’s semi-inverse and improved

(G^{'} / G)

[...] Read more.

We consider in this study the (3+1)-dimensional stochastic potential Yu–Toda–Sasa–Fukuyama with conformable derivative (SPYTSFE-CD) forced by white noise. For different kind of solutions of SPYTSFE-CD, including hyperbolic, rational, trigonometric and function, we use He’s semi-inverse and improved

(G^{'} / G)

-expansion methods. Because it investigates solitons and nonlinear waves in dispersive media, plasma physics and fluid dynamics, the potential Yu–Toda–Sasa–Fukuyama theory may explain many intriguing scientific phenomena. We provide numerous 2D and 3D figures to address how the white noise destroys the pattern formation of the solutions and stabilizes the solutions of SPYTSFE-CD. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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19 pages, 15686 KiB

Open AccessArticle

Heart Rhythm Analysis Using Nonlinear Oscillators with Duffing-Type Connections

by Rodrigue F. Fonkou and Marcelo A. Savi

Fractal Fract. 2023, 7(8), 592; https://doi.org/10.3390/fractalfract7080592 - 31 Jul 2023

Cited by 2 | Viewed by 2370

Abstract

Heartbeat rhythms are related to a complex dynamical system based on electrical activity of the cardiac cells usually measured by the electrocardiogram (ECG). This paper presents a mathematical model to describe the electrical activity of the heart that consists of three nonlinear oscillators [...] Read more.

Heartbeat rhythms are related to a complex dynamical system based on electrical activity of the cardiac cells usually measured by the electrocardiogram (ECG). This paper presents a mathematical model to describe the electrical activity of the heart that consists of three nonlinear oscillators coupled by delayed Duffing-type connections. Coupling alterations and external stimuli are responsible for different cardiac rhythms. The proposed model is employed to build synthetic ECGs representing a variety of responses including normal and pathological rhythms: ventricular flutter, torsade de pointes, atrial flutter, atrial fibrillation, ventricular fibrillation, polymorphic ventricular tachycardia and supraventricular extrasystole. Moreover, the sinoatrial rhythm variations are described by time-dependent frequency, representing transient disturbances. This kind of situation can represent transitions between different pathological behaviors or between normal and pathological physiologies. In this regard, a nonlinear dynamics perspective is employed to describe cardiac rhythms, being able to represent either normal or pathological behaviors. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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16 pages, 12490 KiB

Open AccessArticle

Multistability and Jump in the Harmonically Excited SD Oscillator

by Zhenhua Wang and Huilin Shang

Fractal Fract. 2023, 7(4), 314; https://doi.org/10.3390/fractalfract7040314 - 6 Apr 2023

Cited by 6 | Viewed by 1393

Abstract

Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. [...] Read more.

Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. Then, their stability and conditions for local bifurcation are discussed. Furthermore, the point mapping method is utilized to depict the fractal basins of attraction of the attractors intuitively. Complex hidden attractors, such as period-3 responses and chaos, are found. It follows that jumps among multiple attractors can be easily triggered by an increase in the excitation level or a small disturbance of the initial condition. The results offer an opportunity for a more comprehensive understanding and better utilization of the multistability characteristics of the SD oscillator. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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14 pages, 4114 KiB

Open AccessArticle

A Daisyworld Ecological Parable Including the Revenge of Gaia and Greenhouse Effect

by Marcelo A. Savi and Flavio M. Viola

Fractal Fract. 2023, 7(2), 190; https://doi.org/10.3390/fractalfract7020190 - 14 Feb 2023

Viewed by 2018

Abstract

The Daisyworld model illustrates the concept of biological homeostasis in the global environment by establishing a connection between the biota and environment, resulting in a single intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy populations whereas temperature [...] Read more.

The Daisyworld model illustrates the concept of biological homeostasis in the global environment by establishing a connection between the biota and environment, resulting in a single intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy populations whereas temperature represents the environment, establishing a population dynamics model to represent life–environment ecological interactions. The recent occurrence of extreme weather events due to climate change and the critical crises brought on by the COVID-19 pandemic are strengthening the arguments for the revenge of Gaia, a term used to describe the protective response of the global biota-environment system. This paper presents a novel Daisyworld parable to describe ecological life–environment interactions including the revenge of Gaia and the greenhouse effect. The revenge of Gaia refers to a change in the interplay between life and environment, characterized by the Gaia state that establishes the life-environment state of balance and harmony. This results in reaction effects that impact the planet’s fertile regions. On the other hand, the greenhouse effect is incorporated through the description of the interactions of greenhouse gases with the planet, altering its albedo. Numerical simulations are performed using a nonlinear dynamics perspective, showing different ecological scenarios. An investigation of the system reversibility is carried out together with critical life–environment interactions. This parable provides a qualitative description that can be useful to evaluate ecological scenarios. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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16 pages, 6098 KiB

Open AccessArticle

New Complex Wave Solutions and Diverse Wave Structures of the (2+1)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation

by Guojiang Wu and Yong Guo

Fractal Fract. 2023, 7(2), 170; https://doi.org/10.3390/fractalfract7020170 - 7 Feb 2023

Cited by 11 | Viewed by 1747

Abstract

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new [...] Read more.

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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18 pages, 4564 KiB

Open AccessArticle

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

by Guojiang Wu and Yong Guo

Fractal Fract. 2023, 7(1), 75; https://doi.org/10.3390/fractalfract7010075 - 9 Jan 2023

Cited by 4 | Viewed by 1645

Abstract

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In [...] Read more.

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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22 pages, 2107 KiB

Open AccessArticle

Regime-Switching Fractionally Integrated Asymmetric Power Neural Network Modeling of Nonlinear Contagion for Chaotic Oil and Precious Metal Volatilities

by Melike Bildirici and Özgür Ömer Ersin

Fractal Fract. 2022, 6(12), 703; https://doi.org/10.3390/fractalfract6120703 - 27 Nov 2022

Cited by 2 | Viewed by 1638

Abstract

This paper aims at analyzing nonlinear dependence between fractionally integrated, chaotic precious metal and oil prices and volatilities. With this respect, the Markov regime-switching fractionally integrated asymmetric power versions of generalized autoregressive conditional volatility copula (MS-FIAPGARCH-copula) method are further extended to multi-layer perceptron [...] Read more.

This paper aims at analyzing nonlinear dependence between fractionally integrated, chaotic precious metal and oil prices and volatilities. With this respect, the Markov regime-switching fractionally integrated asymmetric power versions of generalized autoregressive conditional volatility copula (MS-FIAPGARCH-copula) method are further extended to multi-layer perceptron (MLP)-based neural networks copula (MS-FIAPGARCH-MLP-copula). The models are utilized for modeling dependence between daily oil, copper, gold, platinum and silver prices, covering a period from 1 January 1990–25 March 2022. Kolmogorov and Shannon entropy and the largest Lyapunov exponents reveal uncertainty and chaos. Empirical findings show that: i. neural network-augmented nonlinear MS-FIAPGARCH-MLP-copula displayed significant gains in terms of forecasts; ii. asymmetric and nonlinear processes are modeled effectively with the proposed model, iii. important insights are derived with the proposed method, which highlight nonlinear tail dependence. Results suggest, given long memory and chaotic structures, that policy interventions must be kept at lowest levels. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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