2:1:1 Resonance in the Quasi-Periodic Mathieu Equation

We present a small $ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 $ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0      

A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems

We consider the problem of 1:1 resonance in autonomous, timereversible systems. We first present an abstract treatment for n-dimensionalsecond-order systems, and then apply our method to two simplemechanical examples involving follower loads. As the magnitude of the follower load is increased past a criticalvalue, the trivial solution loses stability as the real-valuedfrequencies of the linearized system first coalesce and then splitapart with complex-conjugate values. In Hamiltonian systems this isusually referred to as the Hamiltonian–Hopf bifurcation. Some novelfeatures of our analysis are the direct exploitation of reversibilityand a Liapunov–Schmidt reduction of the second-order (Newtonian)equations of motion, the latter of which requires no complexification.The analysis of the resulting two-parameter, one-dimensionalbifurcation equation yields the possibility that families ofnontrivial periodic solutions may exist for load values in excess of the critical value.     

Steady-state analysis of the Anaerobic Digestion Model No. 1 (ADM1)

The steady-state behavior of the Anaerobic Digestion Model No. 1 (ADM1) with respect to dilution rate and substrate concentration is analyzed in this study. Thereby, up to ten coexisting steady-state solutions are observed under the same operating conditions. The parameter region of a methane producing operation is limited regarding high dilution rates as well as low or high substrate concentrations. The underlying mechanisms causing those limits are investigated in detail, and the common core is identified, namely a positive feedback loop between growth of acetate degraders and acetate itself. The difference lies in the activation mechanisms of this feedback loop, which differs in the three investigated cases. The comparison of the present results with literature studies of simpler two-step models reveals qualitative differences regarding the substrate concentration. Therefore, an alternative simplified model is suggested, which shows qualitatively the same bifurcation behavior as the ADM1 considering variation of the substrate concentration.     

Exact solutions in 1+1 dimensions of the general two-velocity discrete illner model

The Illner model is the most general two-velocity model of the discrete Boltzmann equation. It includes, as particular cases, both the Carleman and the McKean model. Exact solutions in 1+1 dimensions of the general two-velocity discrete Illner model can be studied in a concise way. The conclusions of the precursors need ameliorating. A new type of exact solutions in 1+1 dimensions is obtained. This gives a general method for studying non-trivial exact solutions for the similar discrete Boltzmann equation. Project supported by the National Natural Science Foundation of China (19631060) and the China Post-Doctoral Science Foundation     

Rotation of the apparent vibration plane of a swinging spring at the 1:1:2 resonance

Nonlinear spatial vibrations of a mass point on a weightless elastic suspension (pendulum on a spring) are considered. The frequency of vertical vibrations is assumed to be equal to the doubled swinging frequency (the 1:1:2 resonance). In this case, as numerical calculations and experiments show, the vertical vibrations are unstable, which leads to the vertical vibration energy transfer to the pendulum swinging energy. The vertical vibrations of the mass point decay and, after a certain time period, the pendulum starts swinging in a certain vertical plane. This swinging is also unstable, which results in the reverse energy transfer into the vertical vibration mode. The vertical vibrations are again repeated. But after the second transfer of the vertical vibration energy to the pendulum swinging energy, the apparent plane of vibrations rotates by a certain angle. These effects are described analytically; namely, the energy transfer period, the time variations in the amplitudes of both modes, and the variations in the angle of the apparent vibration plane are determined. An asymptotic solution is also constructed for the mass point trajectory in the orbit elements. In projection on the horizonal plane, the mass point moves in a nearly elliptic trajectory. The ellipse semiaxes slowly vary with time, so that their product remains constant, and the major semiaxis slowly rotates at a constant sectorial velocity. The obtained analytic time dependence of the ellipse semiaxes and the precession angle agree well with the results of numerical calculations.     

温度对PbS

研究了粒径为     

The SESA in the Beginning, Part 1

In recognition of SESA's 25th interest in experimental methods Anniversary in 1968, William M. Murray, co-founder and honorary president of the Society, was asked to prepare a brief history of SESA's origin and early development. This essay was published in the collected edition of the Murray Lectures—1955–1967. In honor of the Society's upcoming 40th Anniversary—to be celebrated at the 1983 Spring Meeting in Cleveland-EIT is reprinting excerpts from this essay.
Part 1, below, describes the organization of the New England Photoelasticity Conference, out of which the modern SESA eventually developed.     

Spatiotemporal dynamics of lump solution to the (1 + 1)-dimensional Benjamin–Ono equation

Nonlinear Dynamics - This paper focuses on the finite-time synchronization problem for a kind of general complex networks with intrinsic time-varying delays and hybrid couplings (i.e., containing...     

Responses of a two degrees-of-freedom system with uncertain parameters in the vicinity of resonance 1:1

Nonlinear Dynamics - Self-excited weakly coupled resonators are an effective way to measure a small biochemical mass in a liquid environment because such resonators can compensate for the viscosity...     

New 1:1:1 periodic solutions in 3-dimensional galactic-type Hamiltonian systems

Applying the averaging theory, we prove the existence of new families of periodic orbits for \(3\) -dimensional type-galactic Hamiltonian systems.     

Stability and approximability of the 𝒫1–𝒫0 element for Stokes equations

In this paper we study the stability and approximability of the ??₁–??₀ element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the ??₁–??₀ element (bilinear/constant on quadrilaterals) and the ??₁–??₀ element. We apply many stability results on the ??₁–??₀ element to the analysis of the ??₁–??₀ element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families. As a byproduct, we also obtain a basis of divergence‐free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence‐free, ??₁ finite element spaces in computation. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic instabilities in {1 1 1} silicon

The phenomena occurring during rapid crack propagation in brittle single crystals was studied by cleaving strip-like silicon specimens along the {1 1 1} low-energy cleavage plane under bending. The experiments reveal phenomena associated with rapid crack propagation in brittle single crystals not previously reported, and new crack path instabilities in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. The experiments demonstrate that crack velocity in single crystals can attain the theoretical limit. No evidence for mirror, mist, and hackle instabilities, typical in amorphous materials, was found. The important role played by the atomistic symmetry of the crystals on controlling and generating the surface instabilities is explained; the importance of the velocity and orientation-dependent cleavage energy is discussed. The surface instabilities are generated to satisfy minimum energy dissipation considerations. These findings necessitate a new approach to the fundamentals of dynamic crack propagation in brittle single crystals.     

Embedded solitons in the $$(2+1)$$(2+1)-dimensional sine-Gordon equation

Nonlinear Dynamics - An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known...     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Subharmonic solutions of the duffing equation with large non-linearity1

The subharmonic solutions of order $\text{1}\text{3}$ of the damped Duffing equation are determined in a suitable parametric form, following the procedure recently developed in [8, 9], and are compared with the results obtained by direct numerical integration of the same equation, carried out with respect to the time with the Runge-Kutta method. It can be deduced that the analytical solution gives satisfactory results in the approximation of the ‘predominantly’ subharmonic solutions of the above equation, even if the non-linearity of the system is very large.     

2:1 Resonance in the delayed nonlinear Mathieu equation

We investigate the dynamics of a delayed nonlinear Mathieu equation: $$\ddot{x}+(\delta+\varepsilon\alpha\,\cos t)x +\varepsilon\gamma x^3=\varepsilon\beta x(t-T)$$ in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of averaging (valid for small ?) is used to obtain a slow flow that is analyzed for stability and bifurcations. We show that the 2:1 instability region associated with parametric excitation can be eliminated for sufficiently large delay amplitudes β, and for appropriately chosen time delays T. We also show that adding delay to an undamped parametrically excited system may introduce effective damping.