This paper presents an innovative numerical scheme designed to simulate variable-order fractional differential operators using power-law, exponential-law, and Mittag-Leffler kernels. Our approach combines the fundamental theorem of fractional calculus with Lagrange polynomial interpolation to improve both accuracy and efficiency. Here we examine a chaotic system of fractional order, characterized by the Caputo derivative, Caputo Fabrizio, and Atangana Baleanu Caputo, with a focus on the implications of constant and variable fractional-order behavior. We generate phase portraits for various fractional orders through a newly developed numerical discretization technique, which also incorporates the Riemann-Liouville fractional integral. For the suggested artificial neural network-based Levenberg-Marquardt (ANN-LM) model, data is generated by numerical computations that include various chaotic dynamics, periodic orbits, and fixed-point conditions. The ANN-LM techniques are used to further train the outcomes that the numerical methods produced. The dataset is used to examine the chaotic behavior of equations that have been modeled, 15% is used for testing, 70% is used for validation, and 15% is used for training. In these sections we also compare numerical solutions with the outcomes that Levenberg-Marquardt neural networks (LM-NNs) are expected to produce. Error histograms, performance assessments, regression analysis, correlation indices, and fitness curves based on the mean squared error (MSE) for the chaotic dynamics, periodic orbits, and fixed points are used to evaluate the ANNs accuracy and efficacy. Tables and graphs representing the chaotic dynamics, periodic orbits, and fixed points are used to further validate the outcomes predicted by the ANN.
The long-term simulation of biological systems at the cellular level presents a significant computational challenge due to pronounced multiscale behavior and severe numerical stiffness. While implicit solvers offer improved stability in solving stiff systems, they can incur prohibitively high computational costs. Alternative conventional methods, such as mass scaling or unit transformation, are often inadequate because model stiffness may arise from multiple causes beyond inertial effects. To address these limitations, this study proposes a model-reduction technique integrated with an adaptive Runge-Kutta solver, which substantially reduces computational time. The effectiveness of the proposed approach is demonstrated by successfully simulating a 14-day adipogenic differentiation process in less than 1 h and 9 min of computational time on a typical desktop computer. Numerical evaluations conducted on a linear spring-mass-damper benchmark, a nonlinear Duffing oscillator, and a detailed human mesenchymal stem cell model show that the proposed approach substantially outperforms conventional scaling techniques and stiff solvers, establishing it as a robust and efficient tool for the long-term simulation of large, multiscale, stiff systems.
The dynamic response of rocking rigid blocks subjected to sinusoidal base excitation exhibits rich nonlinear behavior, with strong sensitivity to initial conditions and external parameters that often leads to chaotic dynamics. Classical analyses have identified bands of forcing frequencies that appear favorable for sustained rocking, whereas others are considered unfavorable due to overturn. This work revisits this assumption by explicitly controlling the initial phase of a second finite-duration sinusoidal pulse. Using an autonomous embedding that enforces phase scheduling and systematic sweeps over the full range phi is an element of[0,2 pi), we show that many frequencies previously deemed unfavorable can sustain bounded oscillations when phase alignment is chosen appropriately. Phase acts not only as a controllable initial condition but also through the implicit time-dependence of the excitation, reshaping the immediate torque balance right after activation. We quantify these effects via long-term response maps in the (tr,phi) plane (reaction time tr) and a complementary dimensionless momentum-impulse metric, revealing windows of stability within the "saw-like" diagrams reported when phase is uncontrolled. The results suggest phase-based control as a simple and effective strategy for mitigating overturn in rocking blocks, refine our understanding of chaotic transitions, and motivate amplitude-phase-frequency codesign for robust stabilization.
Abstract Fractals, with their intricate structures and self-similar patterns, continue to fascinate mathematicians and scientists alike. In this article, we introduce a novel generalization for generating Mandelbrot, Julia, and Multicorn sets by extending the M-iteration process through incorporation of the s-convex combination. We first establish an escape criterion for complex polynomials of the form xk+1+c, which forms the basis of our escape-time algorithms. These algorithms enable the construction of diverse sets of fractals under the proposed iteration scheme, unveiling new structural variations. Additionally, we present graphical representations to visualize these complex sets and conduct a detailed analysis of the interplay between the iteration parameters and two key numerical measures: the average escape time and the nonescaping area index. Our findings reveal highly nonlinear dependencies, shedding new light on the intricate dynamics of fractal evolution.
A physical system exhibiting subcritical Hopf bifurcation followed by a fold bifurcation of the limit cycles presents three operational regimes: a single limit cycle around an unstable equilibrium, a pair of stable and unstable limit cycles around a stable equilibrium, and a globally stable equilibrium. In this study, we investigate whether such an oscillator can be utilized to nonresonantly suppress single-mode instability in structural systems. To this end, we analyze a system comprising a negatively damped linear primary oscillator coupled with a lightweight secondary oscillator capable of subcritical Hopf responses. The secondary oscillator is modeled using a modified van der Pol oscillator exhibiting the stated three regimes of operations. For the single limit cycle regime, prior work (Tandel et al., 2023, "Vibration Stabilization by a Nonresonant Secondary Limit Cycle Oscillator," Nonlinear Dyn., 111(7), pp. 6043-6062) provided a criterion for conditional stabilization, defining the threshold of instability in the primary system for given secondary oscillator parameters. The present study demonstrates that the double limit cycle regime offers conditional stabilization over a significantly larger instability range, leveraging the benefits of the subcritical Hopf dynamics. Finally, the globally stable equilibrium regime offers unconditional stabilization, though its stabilization limit remains equivalent to that using the single limit cycle regime. Hence, a secondary vibration absorber with a stable equilibrium surrounded by an unstable limit cycle which is further surrounded by a larger amplitude stable limit cycle is beneficial as it offers conditional stabilization of an unstable primary system with an arbitrary degree of instability.
