A numerical method of K-S entropy calculation for a strange attractor

Based directly on the original definition of K-S entropy, a new algorithm for calculating K-S entropy from chaotic time series is developed by using some techniques of coding and code operation. The project supported by National Natural Science Foundation of China     

Lyapunov exponents and a strange attractor for the damped nonlinear mathieu equation

Summary A single-degree-of-freedom nonlinear parametrically excited oscillator is considered. Such oscillators provide models for mechanical systems such as shells and plates under periodical load. Chaotic motions and a strange attractor are found to exist applying the theory of Lyapunov exponents. Some difficulties in practical application of the computational procedure for Lyapunov exponents are discussed. Three particular zones for different values of the excitation coefficient are shown to exist, with different type of long-term behavior.

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Lyapunovsche Exponenten und ein seltsamer Attraktor für die nichtlineare Mathieusche Gleichung mit Dämpfung

Übersicht Ein parametererregter nichtlinearer Schwinger mit einem Freiheitsgrad wird untersucht. Solche Schwinger dienen als Modelle für verschiedene mechanische Systeme, z. B. für Platten und Schalen unter periodischer Belastung. Mit Hilfe der Theorie der Lyapunovschen Exponenten ist die Existenz chaotischer Bewegungen und eines seltsamen Attraktors festgestellt worden. Einige Schwierigkeiten bei der Anwendung des numerischen Verfahrens für die Lyapunovschen Exponenten werden diskutiert. Für verschiedene Werte des Erregungskoeffizienten wurden drei wichtige Zonen gefunden, in denen verschiedenes Langzeitverhalten des Systems stattfindet.

     

Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system

Nonlinear Dynamics - We focus on the coexistence of strange nonchaotic attractors (SNAs) and a novel mixed attractor in a periodically driven three-degree-of-freedom vibro-impact system with...     

A novel analytic solution of MHD flow for two classes of visco-elastic fluid over a sheet stretched with non-linearly (quadratic) velocity

The laminar boundary layer flow induced in a quiescent visco-elastic fluid by a permeable stretched flat surface with non-linearly (quadratic) velocity and appropriate wall transpiration under the influence of a magnetic field is investigated. It is shown that the problem permits a complete analytic exponentially decaying solution for the set of continuity and momentum equations with both magnetic field and visco-elasticity influences for two classes of visco-elastic fluid, namely, the second grade and Walters’ liquid B fluids. The effects on both the skin friction parameter α and velocity profiles of various physical parameters such as visco-elasticity, suction/blowing parameter and magnetic parameter are studied. The results for the velocity field are presented through graphs and discussed in detail.     

Wave effects in a stretched band

Gor'kii Branch, Institute of Mechanical Engineering, Academy of Sciences of the USSR. Translated from Prikladnaya Mekhanika, Vol. 27, No. 9, pp. 87–92, September, 1991.     

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

Nonlinear Dynamics - It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A...     

The modulus of particle networks with stretched strands

Expressions are derived which relate the shear modulus G to interaction forces and geometric structure of a particle network with stretched strands. These relations are compared with corresponding expressions given in literature.     

Rate-dependent domain spacing in a stretched NiTi strip

Superelastic fine-grained Nickel–Titanium (NiTi) polycrystalline shape memory alloys under tensile loading deform collectively via the nucleation and growth of macroscopic martensite domains. Recent experiments on a stretched NiTi strip showed that the number of nucleated domains (or the domain spacing) increased (decreased) with increasing applied stretching rate. It is also shown that the rate dependence of the domain formation is due to the coupling between the transfer of the locally released heat and the temperature dependence of the transformation stress. In this paper, a simple one-dimensional model is developed to quantify this effect of thermo-mechanical coupling on the observed domain spacing. Analytical relationship between the domain number, thermo-mechanical properties of the material, heat transfer boundary conditions and the externally applied strain rate is established. It is found that for the case of strong heat convection the domain spacing is inversely proportional to the applied stretching rate, while for the case of weak convection, the domain spacing is dictated by a power-law scaling with exponent ?0.5. The latter theoretical prediction agrees well quantitatively with the experimental data in stagnant air.     

A new chaotic attractor from hybrid optical bistable system

In this work, a new three-dimensional autonomous chaotic system has been introduced by modifying a hybrid optical system. The single quadratic nonlinearity is replaced by a single cubic nonlinearity; the new system can display two 1-scroll chaotic attractors simultaneously or one 2-scroll chaotic attractor. The bifurcation diagram is obtained and Lyapunov spectrum is calculated for the proposed system. The results show that the new system exhibits rich complexity features such as stable, periodic, and chaotic dynamics.     

Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures

This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.     

Heat generation and fracture initiation in a stretched steel plate with a process-induced structural defect

The localization of heat release during deformation of a notched steel plate is studied. It is shown that the source of heat generation in the metal specimen is not the whole region of plastic deformation near the tip of the defect (crack), but only the slip bands occupying the relatively small part of the zone (Chernov-Luders bands), in which the deformation initiates physical and chemical processes. It was found that in the slip bands, the metal temperature increases by several tens of degrees (or more). The deformation of notched specimens is characterized by uneven development of plastic deformation in the volume of the material and a high velocity of heat-wave propagation in the direction of the maximum slip stresses.     

Instantaneous formation of a round hole in a stretched plate

Republic Intercollegiate Computer Center, Kishinev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 11, pp. 90–96, November, 1991.     

Properties of the attractor of a scalar parabolic PDE

We consider a general scalar one-dimensional semilinear parabolic partial differential equation generating a semiflow with an attractor in an adequate state space. Generalizing known results, it is shown that this attractor is the graph of a function over a compact subset of a finite-dimensional subspace of the state space. In addition, we construct an example with a special interest for the geometric or bifurcation theory of this type of parabolic equations.     

Dynamics analysis and Hamilton energy control of a generalized Lorenz system with hidden attractor

Nonlinear Dynamics - Hidden attractor can be found in some dynamic systems. More commonly, it can be excited by the stabilized equilibria, or be generated from the systems without equilibria. The...     

A new chaotic attractor and its robust function projective synchronization

We introduce a simple chaotic system that contains one multiplier and one quadratic term. The system is similar to the generalized Lorenz system but is not topologically equivalent. The properties of the proposed chaotic system are examined by theoretical and numerical analysis. An analog chaotic circuit is implemented that realizes the chaotic system for the verification of its attractor. Furthermore, we propose a robust function projective synchronization using time delay estimation. A numerical simulation of synchronization between the proposed system and the Lorenz system demonstrates that the proposed approach provides fast and robust synchronization even in the presence of unknown parameter variations and disturbances.