It is shown that optimum control of dynamical localization (quantum suppression of classical diff... more It is shown that optimum control of dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic forces having equidistant zeros depends on the impulse transmitted by the external force over half-period rather than on the force amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization.
Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, ho... more Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, however, affect any result of the paper. Indeed, there is no sense in considering the impulse for the case φeff = π/2 (nor for φeff = 3π/2) because, unlike f ∗ φeff=0 (t) (equation (8)), the normalized function f ∗ φeff=π/2 (t) = fφeff=π/2 (t) /[2M(η)] (equation (6)) does not present an ηdependent ‘load’ (constant force) term. Clearly, this is because, unlike the maxima, M (η), and minima, m (η), of fφeff=0 (t) ≡ η cos t + (1 − η) cos (2t), those of fφeff=π/2 (t) ≡ η cos t − (1 − η) sin (2t) are symmetric, i.e., m (η) = −M (η) (compare figures 1(a) and 2(a)). This ultimately comes from the fact that the waveform of fφeff=π/2 (t) fits (for η = 2/3) that of one of the four equivalent expressions of the biharmonic universal excitation
We discuss a general useful theoretical framework to study dynamical localization in ultracold at... more We discuss a general useful theoretical framework to study dynamical localization in ultracold atomic systems confined in periodically shaken optical lattices. Our theory allows to understand some limitations of the usual approach concerning prototypical δ-kicked systems, as well as to explain the experimental results for which finite-time effects cannot be neglected. Specifically, we predict that the strength of dynamical localization reaches a maximum as a function of the width of the pulsatile modulation, whenever its amplitude and period satisfy a given relationship. Additionally, we describe a quite simple scenario for the quantum suppression of classical diffusion, which is confirmed by extensive numerical simulations: The activation of Heisenberg's uncertainty principle giving rise to a drastic reduction of the quantum momentum dispersion if, and only if, the classical dynamics is sufficiently chaotic.
General results concerning maintenance or enhancement of chaos are presented for dissipative syst... more General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.
The aim of the present paper is to show the existence and properties of an exact universal excita... more The aim of the present paper is to show the existence and properties of an exact universal excitation waveform for optimal enhancement of directed ratchet transport (in the sense of the average velocity). This is deduced from the criticality scenario giving rise to ratchet universality, and confirmed by numerical experiments in the context of a driven overdamped Brownian particle subjected to a vibrating periodic potential. While the universality scenario holds regardless of the waveform of the periodic vibratory excitations involved, it is shown that the enhancement of directed ratchet transport is optimal when the impulse transmitted by those excitations (time integral over a halfperiod) is maximum. Additionally, the existence of a frequency-dependent optimal value of the relative amplitude of the two excitations involved is illustrated in the simple case of harmonic excitations.
A theory concerning the emergence and control of chaotic escape from a potential well by means of... more A theory concerning the emergence and control of chaotic escape from a potential well by means of autoresonant excitations is presented in the context of generic, dissipative, and multistable systems. Universal scaling laws relating both the onset and lifetime of transient chaos with the parameters of autoresonant excitations are derived theoretically using vibrational mechanics, Melnikov analysis, and energy-based autoresonance theory. Numerical experiments show that these scaling laws are robust against both the presence of noise and driving reshaping .
We study the effectiveness of locally controlling the impulse transmitted by parametric periodic ... more We study the effectiveness of locally controlling the impulse transmitted by parametric periodic excitations at inducing and suppressing chaos in starlike networks of driven damped pendula, leading to asynchronous chaotic states and equilibria, respectively. We found that the inducing (suppressor) effect of increasing (decreasing) the impulse transmitted by the parametric excitations acting on particular nodes depends strongly on their number and degree of connectivity as well as the coupling strength. Additionally, we provide a theoretical analysis explaining the basic physical mechanisms of the emergence and suppression of chaos as well as the main features of the chaos-control scenario. Our findings constitute proof of the impulse-induced control of chaos in a simple model of complex networks, thus opening the way to its application to real-world networks.
We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the res... more We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the reshaping effect of quasiperiodic forces in nonlinear nonautonomous systems exhibiting strange nonchaotic attractors (SNAs). Specifically, we characterize analytically and numerically some reshaping-induced transitions starting from SNAs in the context of quasiperiodically forced systems. We found similar scenarios of SNAs from the analysis of two representative examples: a quasiperiodically forced damped pendulum and a two-dimensional map. This clearly well-suited and advantageous use of the JEFs, which in their own right lie at the heart of nonlinear physics, may encourage students at intermediate university levels to study them in depth.
The robustness of a chaos-suppressing scenario against potential mismatches is experimentally stu... more The robustness of a chaos-suppressing scenario against potential mismatches is experimentally studied through the universal model of a damped, harmonically driven two-well Duffing oscillator subject to non-harmonic chaos-suppressing excitations. We consider a second order analogous electrical circuit having an extremely simple two-well potential that differs from that of the standard two-well Duffing model, and compare the main theoretical predictions regarding the chaos-suppressing scenario from the latter with experimental results from the former. Our experimental results prove the high robustness of the chaos-suppressing scenario against potential mismatches regardless of the (constant) values of the remaining parameters. Specifically, the predictions of an inverse dependence of the regularization area in the control parameter plane on the impulse of the chaossuppressing excitation as well as of a minimal effective amplitude of the chaos-suppressing excitation when the
World scientific series on nonlinear science, series A, 2005
Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chaco... more Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chacon World Scientific Page 2. Page 3. CONTROL OF HOMOCLINIC CHAOS BV WERK PERIODIC PERTURBATIONS Page 4. ...
Communications in Nonlinear Science and Numerical Simulation, Apr 1, 2020
Abstract We experimentally, numerically, and theoretically characterize the effectiveness of inco... more Abstract We experimentally, numerically, and theoretically characterize the effectiveness of incommensurate excitations at suppressing chaos in damped driven systems. Specifically, we consider an inertial Brownian particle moving in a prototypical two-well potential and subjected to a primary (chaos-inducing) harmonic excitation and a suppressory incommensurategeneric (non-harmonic) excitation. We show that the effective amplitude of the suppressory excitation is minimal when the impulse transmitted by it is near its maximum, while its value is rather insensitive to higher-order convergents of the irrational ratio between the involved driving periods. Remarkably, the number and values of the effective initial phase difference between the two excitations are independent of the impulse while they critically depend on each particular convergent in a complex way involving both the approximate frustration of chaos-inducing homoclinic bifurcations and the maximum survival of relevant spatio-temporal symmetries of the dynamical equation.
International Journal of Bifurcation and Chaos, Dec 1, 1996
This paper studies the effect of continuous and discontinuous time dependent forcings onto dynami... more This paper studies the effect of continuous and discontinuous time dependent forcings onto dynamical systems. We compare these different forcings in the context of laminar chaotic mixing. It is shown that the response of a Hamiltonian two-dimensional system to a time periodic sinusoidal forcing differs qualitatively and quantitatively from the response to a square wave function of the same frequency. Consequently, the mixing efficiency of both types of forcings are different. Also a periodic function of the same shape as that of the velocity of the unperturbed system is tested as a forcing, its mixing efficiency being intermediate.
Intrinsic localized modes, also called discrete breathers, can exist under certain conditions in ... more Intrinsic localized modes, also called discrete breathers, can exist under certain conditions in one-dimensional nonlinear electrical lattices driven by external harmonic excitations. In this work, we have studied experimentally the efectiveness of generic periodic excitations of variable waveform at generating discrete breathers in such lattices. We have found that this generation phenomenon is optimally controlled by the impulse transmitted by the external excitation (time integral over two consecutive zeros), irrespectively of its particular waveform.
The Comment by Quintero et al. does not dispute the central result of our paper [Phys. Rev. E 87,... more The Comment by Quintero et al. does not dispute the central result of our paper [Phys. Rev. E 87, 062114 (2013)] which is a theory explaining the interplay between thermal noise and symmetry breaking in the ratchet transport of a Brownian particle moving on a periodic substrate subjected to a temporal biharmonic excitation γ [η sin (ωt) + α (1 − η) sin (2ωt + ϕ)]. In the Comment, the authors claim, on the sole basis of their numerical simulations for the particular case α = 2, that "there is no such universal force waveform and that the evidence obtained by the authors otherwise is due to their particular choice of parameters." Here we demonstrate by means of theoretical arguments and additional numerical simulations that all the conclusions of our original article are preserved.
The role of the wave form of periodic secondary excitations at controlling (suppressing and enhan... more The role of the wave form of periodic secondary excitations at controlling (suppressing and enhancing) escape from a potential well is investigated. We demonstrate analytically (by Melnikov analysis) and numerically that a judicious choice of the excitation's wave form greatly improves the effectiveness of the escape-controlling excitations while keeping their amplitude and period fixed. These predictions are confirmed by an energy-based analysis that provides the same optimal values of the escape-controlling parameters. The example of a dissipative Helmholtz oscillator is used to illustrate the accuracy of these results.
Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences ... more Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences of notes may illustrate ideas such as the sensitivity to initial conditions, and the dynamics and chaotic behaviour connected with fixed-point and limit-cycle attractors. The aim is not music composition, but a first approach to an interdisciplinary tool suitable for a single session class at preuniversity
We uncover and characterize different chaotic transport scenarios in perfect two-dimensional peri... more We uncover and characterize different chaotic transport scenarios in perfect two-dimensional periodic potentials by controlling the chaotic dynamics of particles subjected to periodic external forces in the absence of a ratchet effect ͑i.e., with no directed transport by symmetry breaking of zero-mean forces͒. After identifying relevant symmetries of the equations of motion, analytical estimates in parameter space for the occurrence of different transport scenarios are provided and confirmed by numerical simulations. These scenarios are highly sensitive to variations of the system's asymmetry parameters, including the eccentricity of the two-dimensional periodic potential and the direction of dc and ac forces, which could be useful for particle sorting purposes in those cases where chaos is unavoidable.
We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of s... more We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of symmetric pulses. The order-chaos threshold when altering solely the width of the pulses is investigated theoretically through Melnikov analysis. We show analytically and numerically that most of the results appear independent of the particular wave form of the pulses provided that the transmitted impulse is the same. By using this property, the stability boundaries of the stationary solutions are determined to first approximation by means of an elliptic harmonic balance method. Finally, the bifurcation behavior at the stability boundaries is determined numerically.
The control of chaos ͑suppression and enhancement͒ of a damped pendulum subjected to two perpendi... more The control of chaos ͑suppression and enhancement͒ of a damped pendulum subjected to two perpendicular periodic excitations of its pivot ͑one chaos inducing and the other chaos controlling͒ is investigated. Analytical ͑Melnikov analysis͒ and numerical ͑Lyapunov exponents͒ results show that the initial phase difference between the two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of the method in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homogeneous regularization is obtained under localized control on a minimal number of pendula. Additionally, we demonstrate the robustness of the control scenario against changes in the coupling function. In particular, synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by considering time-varying couplings instead of stationary couplings.
It is shown that optimum control of dynamical localization (quantum suppression of classical diff... more It is shown that optimum control of dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic forces having equidistant zeros depends on the impulse transmitted by the external force over half-period rather than on the force amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization.
Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, ho... more Equation (7) is incorrect: it should be replaced by I [ f ∗ φeff=π/2 ] (η) = 0. This does not, however, affect any result of the paper. Indeed, there is no sense in considering the impulse for the case φeff = π/2 (nor for φeff = 3π/2) because, unlike f ∗ φeff=0 (t) (equation (8)), the normalized function f ∗ φeff=π/2 (t) = fφeff=π/2 (t) /[2M(η)] (equation (6)) does not present an ηdependent ‘load’ (constant force) term. Clearly, this is because, unlike the maxima, M (η), and minima, m (η), of fφeff=0 (t) ≡ η cos t + (1 − η) cos (2t), those of fφeff=π/2 (t) ≡ η cos t − (1 − η) sin (2t) are symmetric, i.e., m (η) = −M (η) (compare figures 1(a) and 2(a)). This ultimately comes from the fact that the waveform of fφeff=π/2 (t) fits (for η = 2/3) that of one of the four equivalent expressions of the biharmonic universal excitation
We discuss a general useful theoretical framework to study dynamical localization in ultracold at... more We discuss a general useful theoretical framework to study dynamical localization in ultracold atomic systems confined in periodically shaken optical lattices. Our theory allows to understand some limitations of the usual approach concerning prototypical δ-kicked systems, as well as to explain the experimental results for which finite-time effects cannot be neglected. Specifically, we predict that the strength of dynamical localization reaches a maximum as a function of the width of the pulsatile modulation, whenever its amplitude and period satisfy a given relationship. Additionally, we describe a quite simple scenario for the quantum suppression of classical diffusion, which is confirmed by extensive numerical simulations: The activation of Heisenberg's uncertainty principle giving rise to a drastic reduction of the quantum momentum dispersion if, and only if, the classical dynamics is sufficiently chaotic.
General results concerning maintenance or enhancement of chaos are presented for dissipative syst... more General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.
The aim of the present paper is to show the existence and properties of an exact universal excita... more The aim of the present paper is to show the existence and properties of an exact universal excitation waveform for optimal enhancement of directed ratchet transport (in the sense of the average velocity). This is deduced from the criticality scenario giving rise to ratchet universality, and confirmed by numerical experiments in the context of a driven overdamped Brownian particle subjected to a vibrating periodic potential. While the universality scenario holds regardless of the waveform of the periodic vibratory excitations involved, it is shown that the enhancement of directed ratchet transport is optimal when the impulse transmitted by those excitations (time integral over a halfperiod) is maximum. Additionally, the existence of a frequency-dependent optimal value of the relative amplitude of the two excitations involved is illustrated in the simple case of harmonic excitations.
A theory concerning the emergence and control of chaotic escape from a potential well by means of... more A theory concerning the emergence and control of chaotic escape from a potential well by means of autoresonant excitations is presented in the context of generic, dissipative, and multistable systems. Universal scaling laws relating both the onset and lifetime of transient chaos with the parameters of autoresonant excitations are derived theoretically using vibrational mechanics, Melnikov analysis, and energy-based autoresonance theory. Numerical experiments show that these scaling laws are robust against both the presence of noise and driving reshaping .
We study the effectiveness of locally controlling the impulse transmitted by parametric periodic ... more We study the effectiveness of locally controlling the impulse transmitted by parametric periodic excitations at inducing and suppressing chaos in starlike networks of driven damped pendula, leading to asynchronous chaotic states and equilibria, respectively. We found that the inducing (suppressor) effect of increasing (decreasing) the impulse transmitted by the parametric excitations acting on particular nodes depends strongly on their number and degree of connectivity as well as the coupling strength. Additionally, we provide a theoretical analysis explaining the basic physical mechanisms of the emergence and suppression of chaos as well as the main features of the chaos-control scenario. Our findings constitute proof of the impulse-induced control of chaos in a simple model of complex networks, thus opening the way to its application to real-world networks.
We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the res... more We demonstrate the effectiveness of Jacobian elliptic functions (JEFs) for inquiring into the reshaping effect of quasiperiodic forces in nonlinear nonautonomous systems exhibiting strange nonchaotic attractors (SNAs). Specifically, we characterize analytically and numerically some reshaping-induced transitions starting from SNAs in the context of quasiperiodically forced systems. We found similar scenarios of SNAs from the analysis of two representative examples: a quasiperiodically forced damped pendulum and a two-dimensional map. This clearly well-suited and advantageous use of the JEFs, which in their own right lie at the heart of nonlinear physics, may encourage students at intermediate university levels to study them in depth.
The robustness of a chaos-suppressing scenario against potential mismatches is experimentally stu... more The robustness of a chaos-suppressing scenario against potential mismatches is experimentally studied through the universal model of a damped, harmonically driven two-well Duffing oscillator subject to non-harmonic chaos-suppressing excitations. We consider a second order analogous electrical circuit having an extremely simple two-well potential that differs from that of the standard two-well Duffing model, and compare the main theoretical predictions regarding the chaos-suppressing scenario from the latter with experimental results from the former. Our experimental results prove the high robustness of the chaos-suppressing scenario against potential mismatches regardless of the (constant) values of the remaining parameters. Specifically, the predictions of an inverse dependence of the regularization area in the control parameter plane on the impulse of the chaossuppressing excitation as well as of a minimal effective amplitude of the chaos-suppressing excitation when the
World scientific series on nonlinear science, series A, 2005
Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chaco... more Page 1. WORLD SCIENTIFIC SERIES ON ONLINEAR SCIENC Series Editor: Leon O. Chua HIKK Ricardo Chacon World Scientific Page 2. Page 3. CONTROL OF HOMOCLINIC CHAOS BV WERK PERIODIC PERTURBATIONS Page 4. ...
Communications in Nonlinear Science and Numerical Simulation, Apr 1, 2020
Abstract We experimentally, numerically, and theoretically characterize the effectiveness of inco... more Abstract We experimentally, numerically, and theoretically characterize the effectiveness of incommensurate excitations at suppressing chaos in damped driven systems. Specifically, we consider an inertial Brownian particle moving in a prototypical two-well potential and subjected to a primary (chaos-inducing) harmonic excitation and a suppressory incommensurategeneric (non-harmonic) excitation. We show that the effective amplitude of the suppressory excitation is minimal when the impulse transmitted by it is near its maximum, while its value is rather insensitive to higher-order convergents of the irrational ratio between the involved driving periods. Remarkably, the number and values of the effective initial phase difference between the two excitations are independent of the impulse while they critically depend on each particular convergent in a complex way involving both the approximate frustration of chaos-inducing homoclinic bifurcations and the maximum survival of relevant spatio-temporal symmetries of the dynamical equation.
International Journal of Bifurcation and Chaos, Dec 1, 1996
This paper studies the effect of continuous and discontinuous time dependent forcings onto dynami... more This paper studies the effect of continuous and discontinuous time dependent forcings onto dynamical systems. We compare these different forcings in the context of laminar chaotic mixing. It is shown that the response of a Hamiltonian two-dimensional system to a time periodic sinusoidal forcing differs qualitatively and quantitatively from the response to a square wave function of the same frequency. Consequently, the mixing efficiency of both types of forcings are different. Also a periodic function of the same shape as that of the velocity of the unperturbed system is tested as a forcing, its mixing efficiency being intermediate.
Intrinsic localized modes, also called discrete breathers, can exist under certain conditions in ... more Intrinsic localized modes, also called discrete breathers, can exist under certain conditions in one-dimensional nonlinear electrical lattices driven by external harmonic excitations. In this work, we have studied experimentally the efectiveness of generic periodic excitations of variable waveform at generating discrete breathers in such lattices. We have found that this generation phenomenon is optimally controlled by the impulse transmitted by the external excitation (time integral over two consecutive zeros), irrespectively of its particular waveform.
The Comment by Quintero et al. does not dispute the central result of our paper [Phys. Rev. E 87,... more The Comment by Quintero et al. does not dispute the central result of our paper [Phys. Rev. E 87, 062114 (2013)] which is a theory explaining the interplay between thermal noise and symmetry breaking in the ratchet transport of a Brownian particle moving on a periodic substrate subjected to a temporal biharmonic excitation γ [η sin (ωt) + α (1 − η) sin (2ωt + ϕ)]. In the Comment, the authors claim, on the sole basis of their numerical simulations for the particular case α = 2, that "there is no such universal force waveform and that the evidence obtained by the authors otherwise is due to their particular choice of parameters." Here we demonstrate by means of theoretical arguments and additional numerical simulations that all the conclusions of our original article are preserved.
The role of the wave form of periodic secondary excitations at controlling (suppressing and enhan... more The role of the wave form of periodic secondary excitations at controlling (suppressing and enhancing) escape from a potential well is investigated. We demonstrate analytically (by Melnikov analysis) and numerically that a judicious choice of the excitation's wave form greatly improves the effectiveness of the escape-controlling excitations while keeping their amplitude and period fixed. These predictions are confirmed by an energy-based analysis that provides the same optimal values of the escape-controlling parameters. The example of a dissipative Helmholtz oscillator is used to illustrate the accuracy of these results.
Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences ... more Music is presented as a setting for teaching nonlinear dynamics, showing how different sequences of notes may illustrate ideas such as the sensitivity to initial conditions, and the dynamics and chaotic behaviour connected with fixed-point and limit-cycle attractors. The aim is not music composition, but a first approach to an interdisciplinary tool suitable for a single session class at preuniversity
We uncover and characterize different chaotic transport scenarios in perfect two-dimensional peri... more We uncover and characterize different chaotic transport scenarios in perfect two-dimensional periodic potentials by controlling the chaotic dynamics of particles subjected to periodic external forces in the absence of a ratchet effect ͑i.e., with no directed transport by symmetry breaking of zero-mean forces͒. After identifying relevant symmetries of the equations of motion, analytical estimates in parameter space for the occurrence of different transport scenarios are provided and confirmed by numerical simulations. These scenarios are highly sensitive to variations of the system's asymmetry parameters, including the eccentricity of the two-dimensional periodic potential and the direction of dc and ac forces, which could be useful for particle sorting purposes in those cases where chaos is unavoidable.
We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of s... more We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of symmetric pulses. The order-chaos threshold when altering solely the width of the pulses is investigated theoretically through Melnikov analysis. We show analytically and numerically that most of the results appear independent of the particular wave form of the pulses provided that the transmitted impulse is the same. By using this property, the stability boundaries of the stationary solutions are determined to first approximation by means of an elliptic harmonic balance method. Finally, the bifurcation behavior at the stability boundaries is determined numerically.
The control of chaos ͑suppression and enhancement͒ of a damped pendulum subjected to two perpendi... more The control of chaos ͑suppression and enhancement͒ of a damped pendulum subjected to two perpendicular periodic excitations of its pivot ͑one chaos inducing and the other chaos controlling͒ is investigated. Analytical ͑Melnikov analysis͒ and numerical ͑Lyapunov exponents͒ results show that the initial phase difference between the two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of the method in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homogeneous regularization is obtained under localized control on a minimal number of pendula. Additionally, we demonstrate the robustness of the control scenario against changes in the coupling function. In particular, synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by considering time-varying couplings instead of stationary couplings.
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Papers by Ricardo Chacón