Topological classification of periodic orbits in Lorenz system

We systematically investigate the periodic orbits of the Lorenz flow up to certain topological length. As an alternative to Poincar6 section map analysis, we propose a new approach for establishing one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is stable numerically for cycle searching, and two orbital fragments can be used as basic building blocks for initialization. The topological classification based on the entire orbital structure is revealed to be effective. The deformation of periodic orbits with the change of parameters provides a chart to the periods of cycles. The current research may provide a methodology for finding and systematically classifying periodic orbits in other similar chaotic flows.

Stabilizing the Lorenz Flows Using a Closed Loop Quotient Controller

In this study, we introduce a closed loop quotient controller into the three-dimensional Lorenz system. We then compute the equilibrium points and analyze their local stability. We use several examples to illustrate how cross-sections of the basins of attraction for the equilibrium points look for various parameter values. We then provided numerical evidence that with the controller, the controlled Lorenz system cannot exhibit chaos if the equilibrium points are locally stable.

Applying Linear Controls to Chaotic Continuous Dynamical Systems

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.

Software Package Evaluation for Lyapunov Exponent and Others Features of Signals Evaluating Condition Monitoring Performance of Nonlinear Dynamic Systems

Efficient use of industrial equipment, increase its availability, safety and economic issues spur strong research on maintenance programs based on their operating conditions. Machines normally operate in a linear range, but when malfunctions occur, nonlinear behavior might set in. By studying and comparing five nonlinear features, which listed in decreasing order by their damage detection capability are： LLE （largest Lyapunov exponent）, embedded dimension, Kappa determinism, time delay and cross error values; i.e., LLE performs best. Using somewhat similar ideas from Chaos control, i.e., vary the ＂mass imbalance＂ forcing parameters, we aim to stabilize the Lorenz equation. Quite interestingly, for certain imbalance excitation values, the system is stabilized. The previous even when paradigmatically chaotic parameters for Lorenz system are used （plus our forcing terms）. This quasi-control approach is validated studying signals obtained from the previously mentioned lab test. Finally, it is concluded that analyzing and comparing nonlinear features extracted from baseline vs. malfunction condition （test acquired）, one might increase the efficiency and the performance of machine condition monitoring.

MAGNETIC-ELASTIC BUCKLING OF A THIN CURRENT CARRYING PLATE SIMPLY SUPPORTED AT THREE EDGES

The magnetic-elasticity buckling problem of a current plate under the action of a mechanical load in a magnetic field was studied by using the Mathieu function. According to the magnetic-elasticity non-linear kinetic equation, physical equations, geometric equations, the expression for Lorenz force and the electrical dynamic equation, the magnetic-elasticity dynamic buckling equation is derived. The equation is changed into a standard form of the Mathieu equation using Galerkin＇s method. Thus, the buckling problem can be solved with a Mathieu equation. The criterion equation of the buckling problem also has been obtained by discussing the eigenvalue relation of the coefficients 2 and r/ in the Mathieu equation. As an example, a thin plate simply supported at three edges is solved here. Its magnetic-elasticity dynamic buckling equation and the relation curves of the instability state with variations in some parameters are also shown in this paper. The conclusions show that the electrical magnetic forces may be controlled by changing the parameters of the current or the magnetic field so that the aim of controlling the deformation, stress, strain and stability of the current carrying plate is achieved.

A Probabilistic Approach of the Poincaré-Bendixon Problem in Rd

We present how a probabilistic model can describe the asymptotic behavior of the iterations, with applications for ODE and approach of the Poincaré- Bendixon’s problem in Rd.

Comparison of Different Order Adams-Bashforth Methods in an Atmospheric General Circulation Model

The Asselin-Robert time As an attractive alternative filter used in the leaptYog scheme does degrade the accuracy of calculations. to leapfrog time differencing, the second-order Adams-Bashforth method is not subject to time splitting instability and keeps excellent calculation accuracy. A second-order Adams- Bashforth model has been developed, which represents better stability, excellent convergence and improved simulation of prognostic variables. Based on these results, the higher-order Adams-Bashforth methods are developed on the basis of NCAR （National Center for Atmospheric Research） CAM 3.1 （Community Atmosphere Model 3.1） and the characteristics of dynamical cores are analyzed in this paper. By using Lorenz nonlinear convective equations, the filtered leapfrog scheme shows an excellent pattern for eliminating 2At wave solutions after 20 steps but represents less computational solution accuracy. The fourth-order Adams- Bashforth method is closely converged to the exact solution and provides a reference against which other methods may be compared. Thus, the Adams-Bashforth methods produce more accurate and convergent solution with differencing order increasing. The Held-Suarez idealized test is carried out to demonstrate that all methods have similar climate states to the results of many other global models for long-term integration. Besides, higher-order methods perform better in mass conservation and exhibit improvement in simulating tropospheric westerly jets, which is likely equivalent to the advantages of increasing horizontal resolutions. Based on the idealized baroclinic wave test, a better capability of the higher-order method in maintaining simulation stability is convinced. Furthermore, after the baroclinic wave is triggered through overlaying the steady-state initial conditions with the zonal perturbation, the higher-order method has a better ability in the simulation of baroclinic wave perturbation.