Controlling Strange Attractor in Dynamics

A nonlinear system which exhibits a strange attractor is considered, with the goal of illustrating how to control the chaotic dynamical system and how to obtain a desired attracting periodic orbit by the OGY control algorithm.

MODEL SHIFT AND STRANGE ATTRACTOR ON MBIUS STRIP

A new model shift mapping was given in bilateral symbol space. It is topologically conjugate with the traditional shift mapping. Similar to Smale Horseshoe, a model was constructed correspondent to the model shift mapping, i.e., a class of mapping on Mbius strip was given. Its attractors' structure and dynamical behaviour were described.

Strange Attractors in the Resonance Interaction of Two-State Atoms with Single-Mode Laser Field

The resonance interaction of two-state atoms with single mode field is described theoretically by using the semi-classical theory and Jaynes-Cummings model. The nonlinear characteristics of this system are calculated by using FFT and Runge-Kutta methods. The chaotic strange attractors in this system are obtained from the numerical results.

SINAI BOWEN RUELLE MEASURE AND THE STRUCTURE OF STRANGE ATTRACTORS IN THE LAUWERIER MAP

In this paper,the Lauwerier map F a,b (x,y)=(bx(1-2y)+y,ay(1-y)) is considered for a=4 . This map possesses a nontrivial topologically transitive attractor Λ which is the closure of the unstable set of some hyperbolic fixed point. Periodic points are dense in Λ and all of them are hyperbolic with eigenvalues uniformly bounded away from 1 in norm. Moreover,any two periodic points are heteroclinically related (transversal intersection of their stable and unstable sets). The Sinai Bowen Ruelle measure supported on the attractor is constructed and its properties are studied.

The Abundance of Strange Attractor in the Families of Nearby Singular Diffeomorphism

In this paper,we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space,such that for each point of the set the corresponding diffeomorphism possesses strange attractor.For some families of one-dimensional mapping satisfying certain transversality condition,we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor.Furthermore,we study the dynamics of this type of strange attractor.

STRANGE ATTRACTORS IN SYMMETRIC UNFOLDINGS OF A SINGULARITY WITH THREE-FOLD ZERO EIGENVALUE

In this paper,we study the Sil’nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the normal form,the blow-up and the generalized Mel’nikov methods of heteroclinic orbits to two hyperbolic or nonhyperbolic equilibria in a high-dimensional space.

Strange Nonchaotic Attractors in a Quasiperiodically Excited Slender Rigid Rocking Block with Two Frequencies

In this paper,we investigate the existence of strange nonchaotic attractors(SNAs)in a slender rigid rocking block under quasi-periodic forcing with two frequencies.We find that an SNA can exist between a quasi-periodic attractor and a chaotic attractor,or between two chaotic attractors.In particular,we demonstrate that a torus doubling bifurcation of a quasi-periodic attractor can result in SNAs via the fractal route before transforming into chaotic attractors.This phenomenon is rarely reported in quasiperiodically forced discontinuous differential equations and vibro-impact systems.The properties of SNAs are verified by the Lyapunov exponent,rational approximation,phase sensitivity,power spectrum,and separation of nearby trajectories.

A New Seven-modes Truncation of the Plane Incompressible Navier-Stokes EquationsA New Seven-modes Truncation of the Plane Incompressible Navier-Stokes Equations

A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.

Analysis and confirmation of fixed points in logistic mapping digital-flow chaos

The fixed points in logistic mapping digital-flow chaos strange attractor arestudied in detail. When k=n in logistic equation, there exist no more than 2n fixed points, whichare deduced and proved theoretically. Three corollaries about the fixed points of logistic mappingare proposed and proved respectively. These theorem and corollaries provide a theoretical basis forchoosing parameter of chaotic sequences in chaotic secure communication and chaotic digitalwatermarking. And they are testified by simulation.

DYNAMICAL BEHAVIOR OF VISCOELASTIC CYLINDRICAL SHELLS UNDER AXIAL PRESSURES

The hypotheses of the Karman-Donnell theory of thin shells with large deflections and the Boltzmann laws for isotropic linear, viscoelastic materials, the constitutive equations of shallow shells are first derived. Then the governing equations for the deflection and stress function are formulated by using the procedure similar to establishing the Karman equations of elastic thin plates. Introducing proper assumptions, an approximate theory for viscoelastic cylindrical shells under axial pressures can be obtained. Finally, the dynamical behavior is studied in detail by using several numerical methods. Dynamical properties, such ns, hyperchaos, chaos, strange attractor, limit cycle etc., are discovered.

ON PROPERTIES OF HYPERCHAOS: CASE STUDY

Some properties of hyperchaos are exploited by studying both uncoupled and coupled CML. In addition to usual properties of chaotic strange attractors, there are other interesting properties, such as: the number of unstable periodic points embedded in the strange attractor increases dramatically increasing and a large number of low-dimensional chaotic invariant sets are contained in the strange attractor. These properties may be useful for regarding the edge of chaos as the origin of complexity of dynamical systems.

Dynamic Stability of Viscoelastic Plates with Finite Deformation and Shear Effects

Based on Reddy's theory of plates with higher order shear deformations and the Boltzmann superposition principles, the governing equations were established for dynamic stability of viscoelastic plates with finite deformations taking account of shear effects. The Galerkin method was applied to simplify the set of equations. The numerical methods in nonlinear dynamics were used to solve the simplified system. It could be seen that there are plenty of dynamic properties for this kind of viscoelastic plates under transverse harmonic loads. The influences of the transverse shear deformations and material parameter on the dynamic behavior of nonlinear viscoelastic plates were investigated.

A Nine-modes Truncation of the Plane Incompressible Navier-Stokes Equations

In this paper a nine-modes truncation of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. The stationary solutions, the existence of attractor and the global stability of the equations are firmly proved. What is more, that the force f acts on the mode ks and k7 respectively produces two systems, which lead to a much richer and varied phenomenon. Numerical simulation is given at last, which shows a stochastic behavior approached through an involved sequence of bifurcations.

A Five-Mode System of the Navier-Stokes Equations on a Torus

A five-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented;the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some basic dynamical behavior of the new chaos system are revealed.

Dynamical behaviour of Liu system with time delayed feedback

This paper investigates the dynamical behaviour of the Liu system with time delayed feedback. Two typical situations are considered and the effect of time-delay parameter on the dynamics of the system is discussed. It is shown that the Liu system with time delayed feedback may exhibit interesting and extremely rich dynamical behaviour. The evolution of the dynamics is shown to be complex with varying time-delay parameter. Moreover, the strange attractor like ‘wormhole＇ is detected via numerical simulations.

Chaos in a Damped Power System

The analytical investigation of a damped power system has been made based on Holmes and Marsden's theorems. The condition for the occurrence of chaotic motions in the system is that the damping factor has to possess negative values. The numerical simulation results concur very well with theoretical predictions.

Dynamic Properties of Viscoelastic Open Shallow Shells

On the basis of the Kármán Donnell theory of thin shells with large deflections and the Boltzmann laws for linear viscoelastic materials, the mathematical model for viscoelastic open shallow shells was formulated. By using the Galerkin average method, the original integro partial differential dynamic system was simplified as a integro ordinary differential dynamic system, which can be transformed into a ordinary differential dynamic system by introducing new variables. The dynamical behavior was studied by some classical methods. Dynamical properties, such as, chaos, strange attractor, limit cycle etc., were discovered.

Scaling Behaviour of Diffusion Limited Aggregation with Linear Seed

We present a computer model of diffusion limited aggregation with linear seed. The clusters with varying linear seed lengths are simulated, and their pattern structure, fractal dimension and multifractal spectrum are obtained. The simulation results show that the linear seed length has little effect on the pattern structure of the aggregation clusters if its length is comparatively shorter. With its increasing, the linear seed length has stronger effects on the pattern structure, while the dimension D f decreases. When the linear seed length is larger, the corresponding pattern structure is cross alike. The larger the linear seed length is, the more obvious the cross-like structure with more particles clustering at the two ends of the linear seed and along the vertical direction to the centre of the linear seed. Furthermore, the multifractal spectra curve becomes lower and the range of singularity narrower. The longer the length of a linear seed is, the less irregular and nonuniform the pattern becomes.

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.

Quasi-periodically Forced Logistic Map with Weak Liouvillean Frequency

Consider a class of quasi-periodically forced logistic maps T×[0,1]■:(θ,x)→(θ+ω,c(θ)x(1-x))(T=R/Z),whereωis an irrational frequency and c(θ)is a specific bimodal function.We prove that under weak Liouvillean condition on frequency,the strange non-chaotic attractor occurs with negative Lyapunov exponent.This extends the result in[Bjerklov,CMP,2009].