Chaotic and fractal maps in higher-order derivative dynamical systems

Frontiers in Physics, Journal Year: 2025, Volume and Issue: 12

Published: Jan. 10, 2025

Hamiltonian maps are considered a class of dynamical systems that hold meticulous properties used to model large number complex systems. When time flows in with two-dimensional degrees freedom, the trajectories phase space can be analyzed within bidimensional surfaces known as Poincaré sections. The Chirikov–Taylor standard map for two canonical variables (momentum and coordinate) is most renewed characterized by family area-preserving single parameter controls degree chaos. In this study, generalization different problems presented discussed. first problem deals higher-order derivative system (up fourth order) since fourth-order characteristic provides possibility chaotic behavior at all scales including nanoscales where high-order derivatives take place nanosystems. second concerns time-dependent δ -kicked rotor fractal dimensions potential due its important implications quantum This study shows dimensional apparently exhibit orbits patterns, spiral patterns comparable Julia set. Moreover, these additional parameters which control Some lead chaos, others patterns.

Language: Английский

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The old Babylonian algorithm: reborn with a bang and its application in non-linear vibration

Frontiers in Applied Mathematics and Statistics, Journal Year: 2025, Volume and Issue: 10

Published: Jan. 15, 2025

The Old Babylonian Algorithm, a remarkable mathematical gem from ancient Mesopotamia (around 1800-1600 BC), has long been subject of fascination for scholars. This algorithm not only represents the advanced intellectual capabilities Babylonians but also holds great relevance in modern times. It is particularly renowned its ingenious method approximating square roots [1].In recent research, an exciting transformation occurred. traditional Algorithm ingeniously modified to solve differential equations, breathing new life into this technique [1]. In paper, we aim reveal how revival led significant role within complex and captivating realm Non-linear Vibration theory, thereby bridging demonstrating timelessness concepts.Nonlinear vibration analysis presents several specific challenges. Strongly nonlinear oscillators, such as one considered study (in form shown equation ( 1)), are characterized by presence higher-order terms. These nonlinearities can lead dynamic behaviors that difficult analyze using methods [3][4][5]. For instance, homotopy perturbation method, while widely used, attempts convert original problem sequence linear subproblems through introduction parameter, yet it may still face difficulties accurately handling highly systems [2][3][4]. variational iteration another commonly employed approach, limitations dealing with intricacies vibrations [5]. He's frequency formulation, originating Chinese algorithm, offers relatively simple effective means treating demonstrated unique value field dynamics straightforward efficient nature [6,7,8]. However, proposed Ref. [1] provides novel perspective potentially more solution strongly oscillator comparison these methods. combines wisdom adaptations better suit context analysis.By exploring application vibration, hope contribute deeper understanding both heritage practical implications contemporary scientific research. research sheds light on historical significance paves way further advancements dynamics.We consider 35 0 u au bu + =(1)This distinguished existence high-order terms, where b constants. initial conditions set )0 uA = ) 00 u= , A amplitude.Traditionally, numerous have adopted address problems.The gained extensive application. endeavors sub-problems introducing parameter -used approach Moreover, formulation [6,7,8], which stems systems. characteristics [9][10][11][12][13].However, apply old suggested [1]:( 2 1 n nn bu uu   -(2)Where  weighting factor. viewpoint contrast methods.To understand derivation let's following step -bystep explanation.Suppose cos t  (3)where frequency. Utilizing obtain )u aA bA At - -(4)By setting 31 = Figure elaborate -depth between approximate exact diverse range values parameters b. vividly showcases performance scope domain. excellent agreement solutions effectively validates effectiveness algorithm. solid evidence theoretically sound practically applicable.( conclusion, emerged promising oscillators. Through combination adaptations, alternative compared techniques.The successful theoretical soundness applicability. solutions, presented study, compelling effectiveness.Looking ahead, findings future studies applications. MEMS (Micro-Electro-Mechanical Systems), could be enhance optimization behavior devices. By modeling analyzing miniature systems, possible improve their performance, reliability, lifespan.Furthermore, exploration algorithm's capabilities. Future investigations focus extending those time -varying or multiple degrees freedom. Additionally, there potential integrating other computational analytical tools, machine learning algorithms identification system optimization.Overall, resurgence promise advancing our ability handle opening up avenues innovation various engineering disciplines.13.

Language: Английский

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Scrutinizing highly nonlinear oscillators using He’s frequency formula

Sound&Vibration, Journal Year: 2025, Volume and Issue: 59(2), P. 2358 - 2358

Published: March 5, 2025

Highly nonlinear oscillators are examined in their capacity to simulate intricate systems engineering, physics, biology, and finance, as well diverse behavior, rendering them essential the development of resilient technological advancement. Therefore, fundamental purpose current work is analyze He’s frequency formula (HFF) get theoretical explanations many types very oscillators. We investigate, both analytical computational, relationship between elastic forces solution a specific oscillator. This oscillator exhibits significant damping. It assumed that required quantity trigonometric functions matches strong ordinary differential equation (ODE) explains motion. The novel approach definitely takes less processing time complex than traditional perturbation methods were widely used this field. method, which essentially giving linearization ODE, known non-perturbative (NPA). procedure produces new similar linear much harmonic scenario. Readers will benefit from an in-depth account NPA. findings validated by numerical examination using Mathematical Software (MS). (NS) tests yielded fairly findings. well-established principle classical trust on Taylor expansions approximate restoring forces, therefore simplifying situation. When NPA used, vulnerability does not present. Furthermore, enables thorough assessment problems’ stability analysis, was possible prior conventional methodology. Consequently, more appropriate responsibility tool for examining approximations extremely MS.

Language: Английский

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Mathematical approach for rapid determination of pull-in displacement in MEMS devices

Frontiers in Physics, Journal Year: 2025, Volume and Issue: 13

Published: April 7, 2025

Introduction Microelectromechanical systems (MEMS) are pivotal in diverse fields such as telecommunications, healthcare, and aerospace. A critical challenge MEMS devices is accurately determining the pull-in displacement voltage, which significantly impacts device performance. Existing methods, including variational iteration method homotopy perturbation method, often fall short providing precise estimations of these parameters. Methods This study introduces a novel mathematical approach that combines physical insights into phenomenon with theory. The begins definition device's model. By uniquely applying principle incorporating custom-designed functional, set equations derived. These transformed an iterative algorithm for calculating displacement, nonlinear terms addressed through approximation techniques tailored to system’s characteristics. Results Validation using specific examples demonstrates method's accuracy voltage. For instance, oscillator case, exact results were achieved computation time 0.015 s. Compared traditional this yields values rather than approximations, showcasing superior precision efficiency. Discussion proposed offers significant advantages, enhanced accuracy, reduced computational time, minimized error accumulation by solving algebraic instead iterating differential equations. It also exhibits robustness variations initial conditions system Limitations include need modifying criterion when formulation unattainable exclusion environmental factors like temperature pressure fluctuations. Future research should focus on refining models incorporate integrating Galerkin technology. Conclusion advances understanding behavior holds substantial potential design optimization across various applications, further driving progression

Language: Английский

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Monitoring the Distance and Velocity of Protons in a Medium for Biomedical Applications Using a Straightforward Mathematical Approach

Sci, Journal Year: 2025, Volume and Issue: 7(2), P. 49 - 49

Published: April 11, 2025

Radiotherapy is one of the major cancer treatments that uses controlled doses ionizing radiation to damage tumor cells. The monitoring charged particles within a medium tremendous importance in radiotherapy. Monte Carlo methods can be used estimate profiles; however, despite their effectiveness, they are computationally expensive, which limits practicality. To simplify analysis, approximate analytical focused on estimating range and velocity function have been previously derived. Previous solutions include non-elementary functions, such as exponential integral with relativistic coordinate transformations, or use regular perturbation accounting for small effects. In this paper, much simpler approach presented assist practitioners field Using proposed method, particles’ velocities calculated exclusively elementary functions. main advantage approach, aside from its straightforward application, suitability velocities. equations derived paper were successfully tested at radiotherapy level, accommodating protons energies up 350 MeV.

Language: Английский

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Modified Lindstedt-Poincare transformation for fractal resonance approach in vibration 2-DOF heterogeneous system

Journal of low frequency noise, vibration and active control, Journal Year: 2025, Volume and Issue: unknown

Published: April 17, 2025

This study looks at the dynamics of a two-degree-of-freedom system subjected to periodic force and describes it with fractal differential derivatives. The applies harmonic equivalent linearized approach (HELA) convert nonlinear equations into linear systems, allowing for derivation analytical solutions extensive numerical comparison. explores use techniques in capturing complex dynamics, particularly resonance scenarios. It proposes novel compares methods, demonstrating high agreement. research also investigates phenomenon, providing visual evidence, enhancing our understanding its significance stability behavior.

Language: Английский

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Modified Ji-Huan He’s Frequency Formulation for Large-Amplitude Electron Plasma Oscillations

Atoms, Journal Year: 2024, Volume and Issue: 12(12), P. 68 - 68

Published: Dec. 12, 2024

This paper examines oscillations governed by the generic nonlinear differential equation u″=ωp021−u∓2β2uγ, where ωp0, β and γ are positive constants. The aforementioned is of particular importance, as it describes electron plasma influenced temperature effects large oscillation amplitudes. Since no analytical solution exists for period in terms β,γ amplitude, accurate approximations derived. A modified He’s approach used to account non-symmetrical around equilibrium position. motion divided into two parts: umin≤u<ueq ueq<u≤umax, umin umax minimum maximum values u, ueq its value. time intervals each part calculated summed find period. proposed method shows remarkable accuracy compared numerical results. most significant result this that can be readily extended strongly oscillations. It also demonstrated same any case segment function f(u) u″+fu=0 (for ueq<u≤umax) approximated a fifth-degree polynomial containing only odd powers.

Language: Английский

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