Special Session 137: Nonlinear Dynamics, Chaos, and Applications: From Fractional Systems to Astrophysical Models

Controlling the dynamics of a superconducting sphere inside superfluid He-II

Manuel Array\`as Universidad Rey Juan Carlos Spain Co-Author(s):    J. L. Trueba, C. Uriarte

Abstract: Since that pioneering time, the use of electromechanical probes to test the quantum features of superfluidity has been the workhorse of ultra--low--temperature research. Superfluids exert forces on objects through added mass, normal--fluid drag, mutual friction with vortices, and emission of excitations (phonons and rotons). Electromechanical resonators such as torsional oscillators, vibrating wires, quartz tuning forks, and other microelectromechanical devices (MEMS), because of changes in frequency, damping, and electrical response, provide very sensitive signatures of superfluid flow, quasiparticles, and broken symmetries. However, most of these probes suffer from boundary effects, as they must be attached to the walls of experimental cells. Moreover, the type of motion they allow one to explore is limited mainly to oscillatory motion. Thus, the search began for a dynamical, non--contact electromechanical probe that would allow one to separate intrinsic mechanical losses from fluid--induced ones. These requirements naturally lead to levitation. A levitated object has no mechanical suspension, so there are no clamping losses. However, controlling the dynamics of a levitated object poses additional challenges that we will analyse.

Fractional damping term in nonlinear oscillators.

Mattia Coccolo Universidad rey Juan Carlos Spain Co-Author(s):    Mattia Coccolo

Abstract: Fractional-order damping provides a compact way to incorporate memory effects into nonlinear dynamical models beyond instantaneous viscous dissipation. In this talk we investigate how introducing a fractional derivative in the dissipative term reshapes both transient evolution and asymptotic dynamics in nonlinear systems, with the fractional order $\alpha$ as a control parameter. Through systematic numerical experiments, we quantify changes in the time required to reach the long-term regime (e.g., convergence to attractors, well-to-well transitions, or escape events) and compare behaviors across parameter ranges that are classically underdamped and overdamped. The results show that memory can qualitatively reorganize the effective damping landscape: regimes that are simple in the integer-order overdamped limit may become structurally richer when $\alpha$ is varied, with altered thresholds and enhanced sensitivity in parameter space. In addition, fractional dissipation can promote resonance-like amplification and frequency-selective responses without modifying the conservative part of the dynamics, providing a new tuning mechanism for nonlinear response. The presented viewpoint is general and can be extended to coupled oscillators and excitable systems.

Predictive-Switching Control of Stochastic Biochemical Oscillators and Toggle Switches with Contractivity Analysis

Christian Fernandez Perez I2SYSBIO-CSIC Spain Co-Author(s):    Christian Fernandez, Manuel Pajaro, Gabor Szederkenyi, Irene Otero-Muras

Abstract: Stochastic biochemical oscillators, such as genetic toggle switches, exhibit complex dynamics due to intrinsic molecular noise and low molecular counts. While the Chemical Master Equation (CME) provides a rigorous probabilistic description, it is often computationally intractable; Partial Integro-Differential Equation (PIDE) models offer a tractable alternative with solid theoretical foundations. We introduce the Predictive-Switching Controller (PSC), a model-based strategy that evaluates system trajectories under a finite set of input configurations and selects the one optimizing a cost functional. This framework can stabilize low-probability states, preserve transient bimodality, and reshape complex distributions. To accelerate high-dimensional computations, a neural network predicts optimal input actions, maintaining the reliability of the model-based approach. A key theoretical contribution is the proof of L1-contractivity of the PIDE dynamics, ensuring that probability distributions under fixed control profiles remain bounded and are robust to variations in initial conditions. We validate PSC on stochastic toggle-switch networks, demonstrating its ability to maintain unstable states and modulate bimodal distributions effectively. These results highlight PSC as a flexible, robust, and computationally efficient method for controlling stochastic biochemical oscillators, with potential applications in synthetic biology and the design of robust genetic circuits.

Some remarks on Melnikov chaos for smooth and piecewise smooth systems

Matteo Franca University of Florence Italy Co-Author(s):    Calamai, A; Pospisil M.

Abstract: It is well known that a smooth autonomous system which has a homoclinic trajectory (i.e. a trajectory converging to a critical point as $t \to \pm \infty$), and subject to a small periodic forcing, may exhibit a chaotic pattern. A motivating example in this context is given by a forced inverted pendulum. Melnikov theory provides a computable sufficient condition for the existence of a transversal intersection between stable and unstable manifolds: in a smooth context this is enough to guarantee the persistence of the homoclinic trajectory and the insurgence of chaos. In this talk we show that, in piecewise smooth system with a transversal homoclinic point, a generic geometrical obstruction forbids chaotic phenomena which are replaced by new bifurcation scenarios. Further, if this obstruction is removed, chaos may arise again. Piecewise smooth systems are motivated by the study of dry friction, state dependent switches, or impacts. In fact we will also show some results new in a smooth context, concerning multiplicity, position, and size of the Cantor set $\aleph$ of initial conditions from which chaos emanates. In particular we will see that, even if the perturbation is $O(\varepsilon)$, we may find infinitely many distinct Cantor set $\aleph$ located in the same $O(\varepsilon^{\nu})$ neighborhood of the critical point, each corresponding to a different pattern, and where $\nu>1$ is as large as we wish. Further the classical requirement that the trajectories stay $O(\epsilon)$ close to the the critical point, might be replaced by staying $O(\varepsilon^{\nu})$ close to the critical point, again $\nu>1$ can be chosen arbitrarily large.

From discrete to continuous modeling and from continuous to discrete modeling in gas-like models in Econophysics

Ricardo Lopez-Ruiz Univ of Zaragoza Spain Co-Author(s):    Ricardo Lopez-Ruiz

Abstract: We explore the two-way path of modeling in Econophysics. From one side, it is possible to go from the discrete model proposed by Draguslescu and Yakovenko (in the 2000`s) to the continuous Z-model proposed and studied by Lopez-Ruiz and collaborators (in the 2010`s). On the other side, it is possible to go from the continuous modified Z-model proposed by Pomeau and Lopez-Ruiz (in the 2015`s) to the discrete model recently proposed and studied by Breton-Fuertes and Lopez-Ruiz (in the 2025`s). The properties and derivations of these models will be displayed and shown in this oral communication.

Modeling Pandemics in an Age-Structured Population

GUSTAVO A Munoz-Fernandez Universidad Complutense de Madrid Spain Co-Author(s):

Abstract: We present a dynamical system that models the long-term impact of a pandemic on a population structured by age. Our analysis focuses on deriving the basic reproduction number to gain qualitative insight into the stability of the disease-free equilibrium states. This quantity, obtained through the next-generation matrix framework, serves as a key threshold parameter that determines whether an infection can invade and persist within the population. The methodology provides a rigorous mathematical foundation for understanding how age-dependent contact patterns, susceptibility, and demographic processes influence disease dynamics. This approach not only clarifies the role of population structure in shaping epidemic trajectories but also offers a systematic way to evaluate potential intervention strategies. Other aspects, such as the effect of vaccination, can also be incorporated and analyzed within this framework.

Forced chaotic scattering

Jesus M Seoane Universidad Rey Juan Carlos Spain Co-Author(s):

Abstract: In this talk, we present a study of the phenomenon of chaotic scattering under periodic forcing. Previous works showed that the frequency value $\omega = 1$ was the optimal value for which particles escaped from the scattering region in the fastest way. Similarly, for the relativistic regime, the study was conducted in the context of a rotating external force where, depending on the value of the external frequency, both fast and slow escapes were observed. Here, we also study quantitative indicators such as SALI or finite-time Lyapunov exponents, but for the case of non-autonomous systems, specifically, those subjected to periodic forcing, which provide local information about the behavior of trajectories as a function of the forcing frequency and amplitude.

Stochastic resonance in time-delayed bistable active particles

Antonio Alejandro Valido Universidad de Las Palmas de Gran Canaria Spain Co-Author(s):

Abstract: We study stochastic resonance in a bistable active model driven by periodic forcing, time-delayed feedback, and both white and colored noise. Using a probability-density approach, linear response theory, and numerical simulations, we analyze the appereance of stochastic resonance. In the small-delay regime, we derive analytical expressions for the stationary probability density, escape rates, and spectral amplification by combining a perturbative treatment of the delayed dynamics with a WKB-like approximation. Our results show that white and colored noise play competing roles: white noise tends to wash out memory effects, whereas colored noise introduces persistence that can reshape barrier crossing and resonance. Time delay also strongly modulates the dynamics by altering the effective stability of the bistable states and the noise-activated transition rates. Numerical simulations of the stochastic delay equations confirm the analytical predictions and reveal a non-monotonic dependence of the resonance on noise strength, delay, and noise correlation time. These results clarify how memory and non-equilibrium fluctuations jointly control bistable stochastic dynamics.

Escaping dynamics of relativistic protons in the Earth`s magnetosphere

Juan C Vallejo Universidad Rey Juan Carlos Spain Co-Author(s):

Abstract: In this talk, we present the dynamical behavior of charged particles under the influence of the Earth`s magnetic field. This work focuses on the motion of the protons in the relativistic regime, analysing the topology of invariant spatial regions in two magnetic field models, namely, the dipolar and Luhmann tail configurations. We compute the distributions of escape and residence times, the exit basins, and the fractal dimensions of their boundaries, as well as a scaling law between the coefficient of the decay law and the energy of the protons. We also present a robust symplectic numerical scheme which is suitable for modeling trajectories where nonlinear effects and the presence of strong sensitivity to initial conditions are present.