Videos by Álvaro García López
This talk can be watched at https://www.youtube.com/watch?v=pfXW27ZT0-E&t=762 . It presents the c... more This talk can be watched at https://www.youtube.com/watch?v=pfXW27ZT0-E&t=762 . It presents the contents of the research works https://link.springer.com/article/10.1007/s11071-020-05928-5 & https://iopscience.iop.org/article/10.1088/1402-4896/abcad2, where electromagnetic pilot waves are derived for extended charged sources in Maxwell's classical electrodynamics. Extended bodies experience self-forces that, according to the retarded potentials, lead to a description of the motion of the "free" particle in terms of time-delayed differential equations. The feedback interaction between Coulombian and radiative fields among different charged parts of the particle makes uniform motion unstable, triggering a self-oscillation, whose frequency corresponds to the zitterbewegung frequency appearing in Dirac's equation. Finally, the self-energy of the particle is computed, which in addition to relativistic rest and kinetic energy, gives rise to the quantum potential. 30 views
Papers by Álvaro García López
Chaos, Solitons and Fractals, 2025
We consider a nonlinear oscillator with state-dependent time-delay that displays a countably infi... more We consider a nonlinear oscillator with state-dependent time-delay that displays a countably infinite number of nested limit cycle attractors, i.e. megastability. In the low-memory regime, the equation reduces to a self-excited nonlinear oscillator and we use averaging methods to analytically show quasilinear increasing amplitude of the megastable spectrum of quantized quasicircular orbits. We further assign a mechanical energy to each orbit using the Lyapunov energy function and obtain a quadratically increasing energy spectrum and (almost) constant frequency spectrum. We demonstrate transitions between different quantized orbits, i.e. different energy levels, by subjecting the system to an external finite-time harmonic driving. In the absence of external driving force, the oscillator asymptotes towards one of the megastable quantized orbits having a fixed average energy. For a large driving amplitude with frequency close to the limit cycle frequency, resonance drives transitions to higher energy levels. Alternatively, for large driving amplitude with frequency slightly detuned from limit-cycle frequency, beating effects can lead to transitions to lower energy levels. Such driven transitions between quantized orbits form a classical analog of quantum jumps. For excitations to higher energy levels, we show amplitude locking where nearby values of driving amplitudes result in the same response amplitude, i.e. the same final higher energy level. We rationalize this effect based on the basins of different limit cycles in phase space. From a practical viewpoint, our work might find applications in physical and engineering system where controlled transitions between several limit cycles of a multistable dynamical system is desired.
arXiv, 2025
We show that work done by the non-conservative forces along a stable limit-cycle attractor of a d... more We show that work done by the non-conservative forces along a stable limit-cycle attractor of a dissipative dynamical system is always equal to zero. Thus, mechanical energy is preserved on average along periodic orbits. This balance between energy gain and energy loss along different phases of the self-sustained oscillation is responsible for the existence of quantized orbits in such systems. Furthermore, we show that the instantaneous preservation of projected phase-space areas along quantized orbits describes the neutral dynamics of the phase, allowing us to derive from this equation the Wilson-Sommerfeld-like quantization condition. We apply our general results to near-Hamiltonian systems, identifying the fixed points of Krylov-Bogoliubov radial equation governing the dynamics of the limit cycles with the zeros of the Melnikov function. Moreover, we relate the instantaneous preservation of the phase-space area along the quantized orbits to the second Krylov-Bogoliubov equation describing the dynamics of the phase. We test the two quantization conditions in the context of hydrodynamic quantum analogs, where a megastable spectra of quantized orbits have recently been discovered. Specifically, we use a generalized pilot-wave model for a walking droplet confined in a harmonic potential, and find a countably infinite set of nested limit cycle attractors representing a classical analog of quantized orbits. We compute the energy spectrum and the eigenfunctions of this self-excited system.
We use gauge fixing to derive Proca equation from Maxwell’s classical electrodynamics in curved s... more We use gauge fixing to derive Proca equation from Maxwell’s classical electrodynamics in curved spacetime. Further restrictions on the gauge yield the Klein-Gordon equation for scalar bosons. The self-coupling of electromagnetic fields through spacetime curvature originates the inertia of wave packets for non-null field solutions, suggesting an electromagnetic origin of mass. We study the weak field limit of these solutions and prove that the electrovacuum can behave as a charged nonlinear optical medium.
Physical Review E, 2025
A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas t... more A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years, classical non-Markovian wave-particle entities that materialize as walking droplets have been shown to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential by displaying few coexisting limit cycle orbits. By considering a truncated-memory stroboscopic pilot-wave model of the system in the low dissipation regime, we obtain a classical harmonic oscillator perturbed by oscillatory nonconservative forces that display countably infinite coexisting limit-cycle states, also known as megastability. Using averaging techniques in the low-memory regime, we derive analytical approximations of the orbital radii, orbital frequency and Lyapunov energy function of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
arXiv (Cornell University), Feb 14, 2022
We study the thermodynamic efficiency of the Malkus-Lorenz waterwheel. For this purpose, we deriv... more We study the thermodynamic efficiency of the Malkus-Lorenz waterwheel. For this purpose, we derive an exact analytical formula that describes the efficiency of this dissipative structure as a function of the phase space variables and the constant parameters of the dynamical system. We show that, generally, as the machine is progressively driven far from thermodynamic equilibrium by increasing its uptake of matter from the environment, it also tends to increase its efficiency. However, sudden drops in the efficiency are found at critical bifurcation points leading to chaotic dynamics. We relate these discontinuous crises in the efficiency to a reduction of the attractor's average value projected along the phase space dimensions that contribute to the rate of entropy generation in the system. In this manner, we provide a thermodynamic criterion that, presumably, governs the evolution of far-from-equilibrium dissipative systems towards their self-assembly and synchronization into increasingly complex networks and structures.
arXiv, 2024
A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas t... more A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years,
classical non-Markovian wave-particle entities that materialize as walking droplets, have been shown
to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential
via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also known as megastability; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
arXiv, 2024
A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas t... more A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years, classical non-Markovian wave-particle entities that materialize as walking droplets, have been shown
to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also known as megastability; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
Chaos, Solitons and Fractals, 2024
A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the... more A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantum-like features emerging in a model of an idealized one-dimensional WPE in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle's position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set.
A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical ... more A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical system with an uncountably infinite continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite discrete energy levels. We consider a class of nonlinear self-sustained oscillators describing a classical active particle in a harmonic potential. These nonlinear oscillators emerge in the low-memory regime of both state-dependent time-delay systems as well as in non-Markovian stroboscopic models of walking droplets. Using averaging techniques, we prove the existence of a countably infinite number of asymptotically stable quantized orbits, i.e. megastability, for this class of self-excited systems. The set of periodic orbits consists of a sequence of nested limit cycle attractors with quasilinear increasing amplitude and alternating stability, yielding smooth basins of attraction. By using the Lyapunov energy function, we estimate the energy spectra of this megastable set of orbits, and perform numerical simulations to confirm the mathematical analysis. Our formalism can be extended to self-excited particles in general confining potentials, resulting in different energy-frequency relations for these dynamical analogs of quantization.
A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical ... more A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical system with an uncountably infinite continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite discrete energy levels. We consider a class of nonlinear self-sustained oscillators describing a classical active particle in a harmonic potential. These nonlinear oscillators emerge in the low-memory regime of both state-dependent time-delay systems as well as in non-Markovian stroboscopic models of walking droplets. Using averaging techniques, we prove the existence of a countably infinite number of asymptotically stable quantized orbits, i.e. megastability, for this class of self-excited systems. The set of periodic orbits consists of a sequence of nested limit cycle attractors with quasilinear increasing amplitude and alternating stability, yielding smooth basins of attraction. By using the Lyapunov energy function, we estimate the energy spectra of this megastable set of orbits, and perform numerical simulations to confirm the mathematical analysis. Our formalism can be extended to self-excited particles in general confining potentials, resulting in different energy-frequency relations for these dynamical analogs of quantization.
A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical ... more A classical particle in a confining potential gives rise to a Hamiltonian conservative dynamical system with an uncountably infinite continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite discrete energy levels. We consider a class of nonlinear self-sustained oscillators describing a classical active particle in a harmonic potential. These nonlinear oscillators emerge in the low-memory regime of both state-dependent time-delay systems as well as in non-Markovian stroboscopic models of walking droplets. Using averaging techniques, we prove the existence of a countably infinite number of asymptotically stable quantized orbits, i.e. megastability, for this class of self-excited systems. The set of periodic orbits consists of a sequence of nested limit cycle attractors with quasilinear increasing amplitude and alternating stability, yielding smooth basins of attraction. By using the Lyapunov energy function, we estimate the energy spectra of this megastable set of orbits, and perform numerical simulations to confirm the mathematical analysis. Our formalism can be extended to self-excited particles in general confining potentials, resulting in different energy-frequency relations for these dynamical analogs of quantization.
arXiv:2401.05616 [nlin.CD], 2024
A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the... more A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantum-like features emerging in a model of an idealized one-dimensional wave-particle entity in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle’s position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set.
arXiv:2401.05616 [nlin.CD], 2024
A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the... more A macroscopic, self-propelled wave-particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantumlike features emerging in a model of an idealized one-dimensional wave-particle entity in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle's position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set.
Communications in Nonlinear Science and Numerical Simulation, 2021
In open Hamiltonian systems, the escape from a bounded region of phase space according to an expo... more In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.
Modelling Cancer Dynamics Using Cellular Automata
Advanced Mathematical Methods in Biosciences and Applications, 2019
Cellular automata and agent-based models have become the cornerstone of the simulation of many co... more Cellular automata and agent-based models have become the cornerstone of the simulation of many complex biological phenomena. More specifically, they are making major breakthroughs in the understanding of cancer development. Besides, these discrete spatio-temporal models can be hybridized with more traditional models based on differential equations, allowing to faithfully represent multiscale open systems. These systems typically consist of many entities that can perform a vast repertoire of actions, which depend on the concentration of substances diffused in their environments, as well as their mutual interaction through different coupling mechanisms. In the present chapter, we use a hybrid cellular automaton model to explore the dynamics of tumor growth in the presence of an immunological response. A mathematical expression is derived, which describes the speed at which a tumor is erased by a population of immune cytotoxic cells, depending on the morphology of the tumors and the intrinsic capacity of the immune cells to detect and destroy their adversaries. Finally, the coevolution of tumor–immune aggregates is simulated and the likelihood of a prolonged tumor mass dormancy mediated by the immune system is discussed.
arXiv, 2023
We have studied the rich dynamics of a damped particle inside an external double-well potential
... more We have studied the rich dynamics of a damped particle inside an external double-well potential
under the influence of state-dependent time-delayed feedback. In certain regions of the parameter
space, we observe multistability with the existence of two different attractors (limit cycle or strange
attractor) with well separated mean Lyapunov energies forming a two-level system. Bifurcation
analysis reveals that, as the effects of the time-delay feedback are enhanced, chaotic transitions
emerge between the two wells of the double-well potential for the attractor corresponding to the
fundamental energy level. By computing the residence time distributions and the scaling laws
near the onset of chaotic transitions, we rationalize this apparent tunneling-like effect in terms of
the crisis-induced intermittency phenomenon. Further, we investigate the first passage times in
this regime and observe the appearance of a Cantor-like fractal set in the initial history space, a
characteristic feature of hyperbolic chaotic scattering. The non-integer value of the uncertainty
dimension indicates that the residence time inside each well is unpredictable. Finally, we demonstrate
the robustness of this tunneling intermittency as a function of the memory parameter by
calculating the largest Lyapunov exponent.
Chaos, Solitons & Fractals
We study the thermodynamic efficiency of the Malkus-Lorenz waterwheel. For this purpose, we deriv... more We study the thermodynamic efficiency of the Malkus-Lorenz waterwheel. For this purpose, we derive an exact analytical formula that describes the efficiency of this dissipative structure as a function of the phase space variables and the constant parameters of the dynamical system. We show that, generally, as the machine is progressively driven far from thermodynamic equilibrium by increasing its uptake of matter from the environment, it also tends to increase its efficiency. However, sudden drops in the efficiency are found at critical bifurcation points leading to chaotic dynamics. We relate these discontinuous crises in the efficiency to a reduction of the attractor's average value projected along the phase space dimensions that contribute to the rate of entropy generation in the system. In this manner, we provide a thermodynamic criterion that, presumably, governs the evolution of far-from-equilibrium dissipative systems towards their self-assembly and synchronization into increasingly complex networks and structures.
Communications in Nonlinear Science and Numerical Simulation, Dec 1, 2021
In open Hamiltonian systems, the escape from a bounded region of phase space according to an expo... more In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.
PLOS ONE, Jun 16, 2016
The fractional cell kill is a mathematical expression describing the rate at which a certain popu... more The fractional cell kill is a mathematical expression describing the rate at which a certain population of cells is reduced to a fraction of itself. We investigate the mathematical function that governs the rate at which a solid tumor is lysed by a cell population of cytotoxic lymphocytes. We do it in the context of enzyme kinetics, using geometrical and analytical arguments. We derive the equations governing the decay of a tumor in the limit in which it is plainly surrounded by immune cells. A cellular automaton is used to test such decay, confirming its validity. Finally, we introduce a modification in the fractional cell kill so that the expected dynamics is attained in the mentioned limit. We also discuss the potential of this new function for non-solid and solid tumors which are infiltrated with lymphocytes.
Communications in Nonlinear Science and Numerical Simulation, Dec 1, 2019
A multicompartment mathematical model is presented with the goal of studying the role of dose-den... more A multicompartment mathematical model is presented with the goal of studying the role of dose-dense protocols in the context of combination cancer chemotherapy. Dose-dense protocols aim at reducing the period between courses of chemotherapy from three to two weeks or less, in order to avoid the regrowth of the tumor during the meantime and achieve maximum cell kill at the end of the treatment. Inspired by clinical trials, we carry out a randomized computational study to systematically compare a variety of protocols using two drugs of different specificity. Our results suggest that cycle specific drugs can be administered at low doses between courses of treatment to arrest the relapse of the tumor. This might be a better strategy than reducing the period between cycles.
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Videos by Álvaro García López
Papers by Álvaro García López
classical non-Markovian wave-particle entities that materialize as walking droplets, have been shown
to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential
via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also known as megastability; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential via limit cycle orbits. However, the number of co-existing quantized states are limited to at most a few limit cycles. By considering a minimal generalized pilot-wave model of the system in the low-memory and low-dissipation regime, we obtain a classical harmonic oscillator perturbed by an oscillatory non-conservative force that displays a countably infinite coexisting limit-cycle states, also known as megastability; thus forming a dynamical analog of infinite quantized states. Using averaging techniques, we derive an analytical approximation of the orbital radii, orbital frequency and Lyapunov energy function, of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.
under the influence of state-dependent time-delayed feedback. In certain regions of the parameter
space, we observe multistability with the existence of two different attractors (limit cycle or strange
attractor) with well separated mean Lyapunov energies forming a two-level system. Bifurcation
analysis reveals that, as the effects of the time-delay feedback are enhanced, chaotic transitions
emerge between the two wells of the double-well potential for the attractor corresponding to the
fundamental energy level. By computing the residence time distributions and the scaling laws
near the onset of chaotic transitions, we rationalize this apparent tunneling-like effect in terms of
the crisis-induced intermittency phenomenon. Further, we investigate the first passage times in
this regime and observe the appearance of a Cantor-like fractal set in the initial history space, a
characteristic feature of hyperbolic chaotic scattering. The non-integer value of the uncertainty
dimension indicates that the residence time inside each well is unpredictable. Finally, we demonstrate
the robustness of this tunneling intermittency as a function of the memory parameter by
calculating the largest Lyapunov exponent.