We uncover and characterize different chaotic transport scenarios on perfect periodic surfaces by controlling the chaotic dynamics of particles subjected to periodic external forces in the absence of a ratchet effect. After identifying relevant {\it symmetries} of chaotic solutions, 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 periodic surface and the direction of dc and ac forces, which could be useful for particle sorting purposes in those cases where chaos is unavoidable.
Partial control is a technique used in systems with transient chaos. The aim of this control method is to avoid the escape of the orbits from a region Q of the phase space where the transient chaotic dynamics takes place. This technique is based on finding a special subset of Q called the safe set. The chaotic orbit can be sustained in the safe set with a minimum amount of control. In this work we develop a control strategy to gradually lead any chaotic orbit in Q to the safe set by using the safety function. With the technique proposed here, the safe set can be converted into a global attract…
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 l…
The properties of long, numerically-determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the…
The presence of the power-law memory is a significant feature of many natural (biological, physical, etc.) and social systems. Continuous and discrete fractional calculus is the instrument to describe the behavior of systems with the power-law memory. The existence of chaotic solutions is an intrinsic property of nonlinear dynamics (regular and fractional). The behavior of fractional systems can be very different from the behavior of the corresponding systems with no memory. Finding periodic points is essential for understanding regular and chaotic dynamics. Fractional systems do not have peri…
Saitama University（埼玉大学）；Kanazawa University（金泽大学）；Japan Science and Technology Agency（日本科学技术振兴机构）；The University of Tokyo（东京大学）
Photonic artificial intelligence has attracted considerable interest in accelerating machine learning; however, the unique optical properties have not been fully utilized for achieving higher-order functionalities. Chaotic itinerancy, with its spontaneous transient dynamics among multiple quasi-attractors, can be employed to realize brain-like functionalities. In this paper, we propose a method for controlling the chaotic itinerancy in a multi-mode semiconductor laser to solve a machine learning task, known as the multi-armed bandit problem, which is fundamental to reinforcement learning. The …
Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators
在这一系列讲座中，我们简要回顾了非线性振子混沌动力学研究的一些最新进展，特别是阻尼型和驱动型。以Duffing、Bonhoven-van der Pol和MLC电路振荡器为例，简要地解释了与这些系统相关的各种分岔和混沌现象。我们使用数值和分析以及模拟仿真方法来研究这些系统。然后，我们指出如何控制的混沌运动可以影响的算法程序，需要最小的扰动。最后简要讨论了如何实现同向演化混沌系统的同步及其在保密通信中的应用。
One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a pre…
In this work we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work we will apply a new approach of the partial control…
One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems including fluid turbulence, while higher-dimensional invariant tori representing quasi-periodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperch…
Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatio-temporally chaotic systems remains challenging as known methods either show poor convergence properties…
In this paper [published in Phys. Rev. E 102, 042210 (2020)], a new method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wavefunctions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wavefunctions is only of the order of the ratio t H /t E , with t H the Heisenberg time and t E the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynami…
