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Register a new userThe study of Lorenz and Rossler strange attractors by means of quantum theory
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Added by
Yurii Ivanovich Bogdanov
Publication date
2014
fields
Physics
and research's language is
English
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We have developed a method for complementing an arbitrary classical dynamical system to a quantum system using the Lorenz and Rossler systems as examples. The Schrodinger equation for the corresponding quantum statistical ensemble is described in terms of the Hamilton-Jacobi formalism. We consider both the original dynamical system in the position space and the conjugate dynamical system corresponding to the momentum space. Such simultaneous consideration of mutually complementary position and momentum frameworks provides a deeper understanding of the nature of chaotic behavior in dynamical systems. We have shown that the new formalism provides a significant simplification of the Lyapunov exponents calculations. From the point of view of quantum optics, the Lorenz and Rossler systems correspond to three modes of a quantized electromagnetic field in a medium with cubic nonlinearity. From the computational point of view, the new formalism provides a basis for the analysis of complex dynamical systems using quantum computers.
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A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $mathfrak{u}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.
In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.
The amplitude-dependent frequency of the oscillations, termed emph{nonisochronicity}, is one of the essential characteristics of nonlinear oscillators. In this paper, the dynamics of the Rossler oscillator in the presence of nonisochronicity is examined. In particular, we explore the appearance of a new fixed point and the emergence of a coexisting limit-cycle and quasiperiodic attractors. We also describe the sequence of bifurcations leading to synchronized, desynchronized attractors and oscillation death states in the coupled Rossler oscillators as a function of the strength of nonisochronicity and coupling parameters. Further, we characterize the multistability of the coexisting attractors by plotting the basins of attraction. Our results open up the possibilities of understanding the emergence of coexisting attractors, and into a qualitative change of the collective states in coupled nonlinear oscillators in the presence of nonisochronicity.
This work presents the continuation of the recent article The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension, published in the Nonlinear Dynamics journal. In this work, in comparison with the results for classical real-valued Lorenz system (henceforward -- Lorenz system), the problem of analytical and numerical identification of the boundary of global stability for the complex-valued Lorenz system (henceforward -- complex Lorenz system) is studied. As in the case of the Lorenz system, to estimate the inner boundary of global stability the possibility of using the mathematical apparatus of Lyapunov functions (namely, the Barbashin-Krasovskii and LaSalle theorems) is demonstrated. For additional analysis of homoclinic bifurcations in complex Lorenz system a special analytical approach by Vladimirov is utilized. To outline the outer boundary of global stability and identify the so-called hidden boundary of global stability, possible birth of hidden attractors and transient chaotic sets is analyzed.
For every $rinmathbb{N}_{geq 2}cup{infty}$, we show that the space of ergodic measures is path connected for $C^r$-generic Lorenz attractors while it is not connected for $C^r$-dense Lorenz attractors. Various properties of the ergodic measure space for Lorenz attractors have been showed. In particular, a $C^r$-connecting lemma ($rgeq2$) for Lorenz attractors also has been proved. In $C^1$-topology, we obtain similar properties for singular hyperbolic attractors in higher dimensions.
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