Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. This paper has investigated the ultimate bound and positively invariant set of a permanent magnet synchronous motor system. We combine the Lyapunov stability theory with the comparison principle method. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set for all the positive values of its parameters σ, γ. In addition, the two-dimensional bound with respect to x ? y are established. Then, it is the two-dimensional estimation about x ? z. Finally, the result is applied to the study of completely chaos synchronization. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme. At the same time, one numerical example illustrating a localization of a chaotic attractor is presented as well. Numerical simulation is consistent with the results of theoretical calculation.     

Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system using a technique combining the generalized Lyapunov function theory and optimization. For the Lorenz–Haken system, we derive a four-dimensional ellipsoidal ultimate bound and positively invariant set. Furthermore, the two-dimensional parabolic ultimate bound with respect to x–z is established. Finally, numerical results to estimate the ultimate bound are also presented for verification. The results obtained in this paper are important and useful in control, synchronization of hyperchaos and their applications.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

统一混沌系统的全局指数吸引集的新结果

研究了参数α∈[1/29,14/173)时,统一混沌系统的全局指数吸引集问题.通过线性变换和广义Lyapunov函数方法,给出了系统最终上界的精确估计.所得结果发展和丰富了现有混沌系统吸引集的结果,并将在混沌控制和同步中得到广泛应用.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

Dynamical behaviors of a new hyperchaotic system

Currently, chaotic systems and chaos‐based applications are commonly used in the engineering fields. One of the main structures used in these applications is chaotic control and synchronization. In this paper, the dynamical behaviors of a new hyperchaotic system are considered. Based on Lyapunov Theorem with differential and integral inequalities, the global exponential attractive sets and positively invariant sets are obtained. Furthermore, the rate of the trajectories is also obtained. The global exponential attractive sets of the system obtained in this paper also offer theoretical support to study chaotic control, chaotic synchronization for this system. Computer simulation results show that the proposed method is effective. Copyright © 2014 John Wiley & Sons, Ltd.     

Bounds of the hyper-chaotic Lorenz–Stenflo system

To estimate the ultimate bound and positively invariant set for a dynamical system is an important but quite challenging task in general. This paper attempts to investigate the ultimate bounds and positively invariant sets of the hyper-chaotic Lorenz–Stenflo (L–S) system, which is based on the optimization method and the comparison principle. A family of ellipsoidal bounds for all the positive parameters values a, b, c, dand a cylindrical bound for a > 0, b > 1, c > 0, d > 0 are derived. Numerical results show the effectiveness and advantage of our methods.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

On the formulation of invariant imbedding problems

The formulation of an invariant imbedding problem from a given linear, two-point boundary-value problem is not unique. In this paper, we illustrate how the formulation of the problem by partitioning the original vectory(z) into [u(z),v(z)], can affect the numerical accuracy. In fact, the partitioning, the choice of theR, O system orS, T system of equations in Scott's method, the location and number of switch points, and the switching procedure, all influence the numerical results and the ease of obtaining solutions. A new method of switching and the appropriate formulas are described, namely, the repeated switching from theR, Q system to theR, Q system of equations or from theS, T system to theS, T system of equations.     

Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp₀|, is found numerically and described analytically (with α to be the normalized recoil frequency and p₀ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion z_out, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion z_in that the probability of detecting any value of z_out in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.     

On synchronization of chaotic systems based on the Thau observer design

In this paper we revisit the Thau observer design and concern its application to the synchronization problem of two Lorenz name related systems in the master-slave formalism. The first one is the Lorenz-Stenflo system possessing a positively invariant ellipsoid while another one is the hyperchaotic Lorenz system possessing a positively invariant cylinder. Information about loci of these invariant domains is applied for the observer design. Further, we present one assertion related to one spectral inequality arisen in the process of assigning stable spectrum to the observer matrix and show its use in the observer design. We demonstrate the efficiency of synchronization schemes for the both of systems with help of numerical simulation.     

On the Partial Sums of the Rearrangements of a Complex Series

Let {z_j} be a set of complex numbers, all with magnitude ? 1, which sum to a real number ε?1. Then the z_j's can be rearranged so that all partial sums are no greater in magnitude than (ε² + 1)^1/2, and this bound is best possible. Bergström showed that if , the best bound is .     

New estimate the bounds for the generalized Lorenz system

The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Cycle Chains and the LPA Model

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.     

A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms

In this paper we give a realization of the Shimura curve for the quaternion algebra over Q with discriminant 6 as a quotient space of the complex upper half plane by the triangle group Δ(3, 6, 6). It is given by the Schwarz map for the Gauss hypergeometric differential equation E(\frac16,\frac13,\frac23){E\left(\frac{1}{6},\frac{1}{3},\frac{2}{3}\right)}. The corresponding abelian surfaces are obtained as isogenous components of the Jacobi varieties of the Picard curves C(s): w³=z(z-1) (z-\frac12 (1-s))(z-\frac12 (1+s)){C(s): w^3=z(z-1)\break \left(z-\frac{1}{2} (1-s)\right)\left(z-\frac{1}{2} (1+s)\right)}.     

Cycles in Newton´s means

In the dynamical system defined by Newton′s means for n complex variables, n≥2 there are invariant, planar curves with (chaotic) dynamics conjugated to the dynamics of z↦z ⁿ on the unit circle in the complex plane. Oblatum 8-III-2000 & 8-XI-2000?Published online: 5 March 2001     

Localization of compact invariant sets and global stability in analysis of one tumor growth model

In this paper, we study some features of global behavior of the four‐dimensional superficial bladder cancer model with Bacillus Calmette‐Guérin (BCG) immunotherapy described by Bunimovich‐Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells. We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor‐free equilibrium point in the positive orthant. Copyright © 2013 John Wiley & Sons, Ltd.     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.