We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteri... more We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteristic feature in several physical situations. In our specific scenario, as well as in the non-fractional case, for large enough excitation amplitudes, all initial conditions are escaping from the potential well. To address this, we incorporate the phase control technique into a parametric term, a feature commonly encountered in real-world situations. In the non-fractional case it has been shown that, a phase difference of ϕ OP T ≈ π, is the optimal value to avoid the escapes of the particles from the potential well. Here, our investigation focuses on understanding when particles escape, considering both the phase difference ϕ and the fractional parameter α as control parameters. Our findings unveil the robustness of phase control, as evidenced by the consistent oscillation of the optimal ϕ value around its non-fractional counterpart when varying the fractional parameter. Additionally, our results underscore the pivotal role of the fractional parameter in governing the proportion of bounded particles, even when utilizing the optimal phase.
The interaction between the fractional order parameter and the damping parameter can play a relev... more The interaction between the fractional order parameter and the damping parameter can play a relevant role for introducing different dynamical behaviors in a physical system. Here, we study the Duffing oscillator with a fractional damping term. Our findings show that for certain values of the fractional order parameter, the damping parameter, and the forcing amplitude high oscillations amplitude can be induced. This phenomenon is due to the appearance of a resonance in the Duffing oscillator only when the damping term is fractional.
International Journal of Bifurcation and Chaos, 2013
Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Par... more Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon-Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.
International Journal of Bifurcation and Chaos, Nov 1, 2015
In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energ... more In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energy harvesting purposes. A system, driven by a low frequency and a high frequency forcing, can give birth to the vibrational resonance phenomenon, when the two forcing amplitudes resonate and a maximum in amplitude is reached. We apply this idea to a bistable oscillator that can convert environmental kinetic energy into electrical energy, that is, an energy harvester. Normally, the VR phenomenon is studied in terms of the forcing amplitudes or of the frequencies, that are not always easy to adjust and change. Here, we study the VR generated by tuning another parameter that is possible to manipulate when the forcing values depend on the environmental conditions. We have investigated the dependence of the maximum response due to the VR for small and large variations in the forcing amplitudes and frequencies. Besides, we have plotted color coded figures in the space of the two forcing amplitudes, in which it is possible to appreciate different patterns in the electrical power generated by the system. These patterns provide useful information on the forcing amplitudes in order to produce the optimal electrical power.
The main purpose of this paper is to study both the underdamped and the overdamped dynamics of th... more The main purpose of this paper is to study both the underdamped and the overdamped dynamics of the nonlinear Helmholtz oscillator with a fractionalorder damping. For that purpose, we use the Grünwald-Letnikov fractional derivative algorithm in order to get the numerical simulations. Here, we investigate the effect of taking the fractional derivative in the dissipative term in function of the parameter α. Our main findings show that the trajectories can remain inside the well or can escape from it depending on α which plays the role of a control parameter. Besides, the parameter α is also relevant for the creation or destruction of chaotic motions. On the other hand, the study of the escape times of the particles from the well, as a result of variations of the initial conditions and the undergoing force F, is reported by the use of visualization techniques such as basins of attraction and bifurcation diagrams, showing a good agreement with previous
Philosophical Transactions of the Royal Society A, Jan 18, 2021
Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillator ar... more Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillator are considered in this paper. We analyze the generation of a certain damping-induced unpredictability, due to the gradual suppression of interwell oscillations. We find the minimal amount of the forcing amplitude and the right forcing frequency to revert the effect of the dissipation, so that the interwell oscillations can be restored, for different time delay values. This is achieved by using the delay-induced resonance, in which the time delay replaces one of the two periodic forcings present in the vibrational resonance. A discussion in terms of the time delay of the critical values of the forcing for which the delay-induced resonance can tame the dissipation effect is finally carried out.
International Journal of Bifurcation and Chaos, Mar 15, 2020
The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term ma... more The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term may be used as an effective enhancer of the oscillations caused by an external forcing maintaining the same frequency. This is possible for the parameters for which the time-delay induces sustained oscillations. Here, we study this type of resonance in the overdamped and underdamped time-delayed Duffing oscillators, and we explore some new features. One of them is the conjugate phenomenon: the oscillations caused by the time-delay may be enhanced by means of the forcing without modifying their frequency. The resonance takes place when the frequency of the oscillations induced by the time-delay matches the ones caused by the forcing and vice versa. This is an interesting result as the nature of both perturbations is different. Even for the parameters for which the time-delay does not induce sustained oscillations, we show that a resonance may appear following a different mechanism.
We consider the nonlinear Duffing oscillator in presence of fractional damping which is character... more We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic in different physical situations. The system is studied with a smaller and larger damping parameter value, that we call the underdamped and overdamped regimes. In both we have studied the relation between the fractional parameter, the amplitude of the oscillations and the times to reach the asymptotic behavior, called asymptotic times. In the overdamped regime, the study shows that, also here, there are oscillations for fractional order derivatives and their amplitudes and asymptotic times can suddenly change for small variations of the fractional parameter. In addition, in this latter regime, a resonant-like behavior can take place for suitable values of the parameters of the system. These results are corroborated by calculating the corresponding Q−factor. We expect that these results can be useful for a better understanding of fractional dynamics and its possible applications as in modeling different kind of materials that normally need complicated damping terms.
We analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic ex... more We analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic external forcing. We show that, for certain values of the delay, the response can be greatly enhanced by a very small forcing amplitude. This phenomenon is related to the presence of a Bogdanov-Takens bifurcation and displays some analogies to other resonance phenomena, but also substantial differences.
We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. F... more We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. For the correct parameter choice the phase space presents two chaotic regions separated by a transient chaotic region in between. One of the chaotic regions is the responsible to give birth to the neuronal bursting regime. Normally, an orbit in this chaotic region cannot pass through the transient chaotic one and reach the other chaotic region. As a consequence the burstings are short in time. Here, we propose a control technique to connect both chaotic regions and allow the neuron to exhibit very long burstings. This control method defines a region Q covering the transient chaotic region where it is possible to find an advantageous set S ⊂ Q through which the orbits can be driven with a minimal control. In addition we show how the set S changes depending on the noise intensity affecting the map, and how the set S can be used in different scenarios of control. I.
We consider the nonlinear Duffing oscillator in presence of fractional damping which is character... more We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic in different physical situations. The system is studied with a smaller and larger damping parameter value, that we call the underdamped and overdamped regimes. In both we have studied the relation between the fractional parameter, the amplitude of the oscillations and the times to reach the asymptotic behavior, called asymptotic times. In the overdamped regime, the study shows that, also here, there are oscillations for fractional order derivatives and their amplitudes and asymptotic times can suddenly change for small variations of the fractional parameter. In addition, in this latter regime, a resonant-like behavior can take place for suitable values of the parameters of the system. These results are corroborated by calculating the corresponding Q−factor. We expect that these results can be useful for a better understanding of fractional dynamics and its possible applications as in modeling different kind of materials that normally need complicated damping terms.
International Journal of Bifurcation and Chaos, 2013
Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Par... more Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon–Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.
We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteri... more We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteristic feature in several physical situations. In our specific scenario, as well as in the non-fractional case, for large enough excitation amplitudes, all initial conditions are escaping from the potential well. To address this, we incorporate the phase control technique into a parametric term, a feature commonly encountered in real-world situations. In the non-fractional case it has been shown that, a phase difference of ϕ OP T ≈ π, is the optimal value to avoid the escapes of the particles from the potential well. Here, our investigation focuses on understanding when particles escape, considering both the phase difference ϕ and the fractional parameter α as control parameters. Our findings unveil the robustness of phase control, as evidenced by the consistent oscillation of the optimal ϕ value around its non-fractional counterpart when varying the fractional parameter. Additionally, our results underscore the pivotal role of the fractional parameter in governing the proportion of bounded particles, even when utilizing the optimal phase.
The interaction between the fractional order parameter and the damping parameter can play a relev... more The interaction between the fractional order parameter and the damping parameter can play a relevant role for introducing different dynamical behaviors in a physical system. Here, we study the Duffing oscillator with a fractional damping term. Our findings show that for certain values of the fractional order parameter, the damping parameter, and the forcing amplitude high oscillations amplitude can be induced. This phenomenon is due to the appearance of a resonance in the Duffing oscillator only when the damping term is fractional.
International Journal of Bifurcation and Chaos, 2013
Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Par... more Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon-Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.
International Journal of Bifurcation and Chaos, Nov 1, 2015
In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energ... more In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energy harvesting purposes. A system, driven by a low frequency and a high frequency forcing, can give birth to the vibrational resonance phenomenon, when the two forcing amplitudes resonate and a maximum in amplitude is reached. We apply this idea to a bistable oscillator that can convert environmental kinetic energy into electrical energy, that is, an energy harvester. Normally, the VR phenomenon is studied in terms of the forcing amplitudes or of the frequencies, that are not always easy to adjust and change. Here, we study the VR generated by tuning another parameter that is possible to manipulate when the forcing values depend on the environmental conditions. We have investigated the dependence of the maximum response due to the VR for small and large variations in the forcing amplitudes and frequencies. Besides, we have plotted color coded figures in the space of the two forcing amplitudes, in which it is possible to appreciate different patterns in the electrical power generated by the system. These patterns provide useful information on the forcing amplitudes in order to produce the optimal electrical power.
The main purpose of this paper is to study both the underdamped and the overdamped dynamics of th... more The main purpose of this paper is to study both the underdamped and the overdamped dynamics of the nonlinear Helmholtz oscillator with a fractionalorder damping. For that purpose, we use the Grünwald-Letnikov fractional derivative algorithm in order to get the numerical simulations. Here, we investigate the effect of taking the fractional derivative in the dissipative term in function of the parameter α. Our main findings show that the trajectories can remain inside the well or can escape from it depending on α which plays the role of a control parameter. Besides, the parameter α is also relevant for the creation or destruction of chaotic motions. On the other hand, the study of the escape times of the particles from the well, as a result of variations of the initial conditions and the undergoing force F, is reported by the use of visualization techniques such as basins of attraction and bifurcation diagrams, showing a good agreement with previous
Philosophical Transactions of the Royal Society A, Jan 18, 2021
Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillator ar... more Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillator are considered in this paper. We analyze the generation of a certain damping-induced unpredictability, due to the gradual suppression of interwell oscillations. We find the minimal amount of the forcing amplitude and the right forcing frequency to revert the effect of the dissipation, so that the interwell oscillations can be restored, for different time delay values. This is achieved by using the delay-induced resonance, in which the time delay replaces one of the two periodic forcings present in the vibrational resonance. A discussion in terms of the time delay of the critical values of the forcing for which the delay-induced resonance can tame the dissipation effect is finally carried out.
International Journal of Bifurcation and Chaos, Mar 15, 2020
The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term ma... more The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term may be used as an effective enhancer of the oscillations caused by an external forcing maintaining the same frequency. This is possible for the parameters for which the time-delay induces sustained oscillations. Here, we study this type of resonance in the overdamped and underdamped time-delayed Duffing oscillators, and we explore some new features. One of them is the conjugate phenomenon: the oscillations caused by the time-delay may be enhanced by means of the forcing without modifying their frequency. The resonance takes place when the frequency of the oscillations induced by the time-delay matches the ones caused by the forcing and vice versa. This is an interesting result as the nature of both perturbations is different. Even for the parameters for which the time-delay does not induce sustained oscillations, we show that a resonance may appear following a different mechanism.
We consider the nonlinear Duffing oscillator in presence of fractional damping which is character... more We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic in different physical situations. The system is studied with a smaller and larger damping parameter value, that we call the underdamped and overdamped regimes. In both we have studied the relation between the fractional parameter, the amplitude of the oscillations and the times to reach the asymptotic behavior, called asymptotic times. In the overdamped regime, the study shows that, also here, there are oscillations for fractional order derivatives and their amplitudes and asymptotic times can suddenly change for small variations of the fractional parameter. In addition, in this latter regime, a resonant-like behavior can take place for suitable values of the parameters of the system. These results are corroborated by calculating the corresponding Q−factor. We expect that these results can be useful for a better understanding of fractional dynamics and its possible applications as in modeling different kind of materials that normally need complicated damping terms.
We analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic ex... more We analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic external forcing. We show that, for certain values of the delay, the response can be greatly enhanced by a very small forcing amplitude. This phenomenon is related to the presence of a Bogdanov-Takens bifurcation and displays some analogies to other resonance phenomena, but also substantial differences.
We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. F... more We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. For the correct parameter choice the phase space presents two chaotic regions separated by a transient chaotic region in between. One of the chaotic regions is the responsible to give birth to the neuronal bursting regime. Normally, an orbit in this chaotic region cannot pass through the transient chaotic one and reach the other chaotic region. As a consequence the burstings are short in time. Here, we propose a control technique to connect both chaotic regions and allow the neuron to exhibit very long burstings. This control method defines a region Q covering the transient chaotic region where it is possible to find an advantageous set S ⊂ Q through which the orbits can be driven with a minimal control. In addition we show how the set S changes depending on the noise intensity affecting the map, and how the set S can be used in different scenarios of control. I.
We consider the nonlinear Duffing oscillator in presence of fractional damping which is character... more We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic in different physical situations. The system is studied with a smaller and larger damping parameter value, that we call the underdamped and overdamped regimes. In both we have studied the relation between the fractional parameter, the amplitude of the oscillations and the times to reach the asymptotic behavior, called asymptotic times. In the overdamped regime, the study shows that, also here, there are oscillations for fractional order derivatives and their amplitudes and asymptotic times can suddenly change for small variations of the fractional parameter. In addition, in this latter regime, a resonant-like behavior can take place for suitable values of the parameters of the system. These results are corroborated by calculating the corresponding Q−factor. We expect that these results can be useful for a better understanding of fractional dynamics and its possible applications as in modeling different kind of materials that normally need complicated damping terms.
International Journal of Bifurcation and Chaos, 2013
Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Par... more Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon–Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.
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Papers by Mattia Coccolo