d2udϕ2=32u2−u (3) We can immediately reproduce the stationary orbit of the photon sphere for the Schwarzschild black hole by settingd2ueqdϕ2=0:req=Rsueq=32Rs. Note, the trivial equilibrium solution foru=0(i.e. at infinite distances from the black hole) will not be discussed ...
\(E_2\equiv (l,0)\), the second axial equilibrium. Existence and the multiplicity of positive equilibria depends upon the shapes and positions of the nontrivial nulclines and the parametric restrictions. If\(E_{3}\equiv (x^*, y^*)\)is positive equilibrium hence\(y^*=\frac{\beta x...
But this equilibrium is unstable: the slightest windfall capital accumulation will start the system off …” Our analysis challenges these assessments. We show that a neoclassical economy may take-off even though the initial capital stock is zero and capital is essential, i.e. the system can ...
The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the ... AD Egidio,A Luongo,A Paolone - 《International Journal of Non Linear Mechanics》 被引量: 108发表: 2007年 Stability analysis of the spat...
Find the nontrivial equilibrium point. {dx} / {dt} = x^2 + x y - 2 x {dy} / {dt} = x y - 2 y^2 - 2 y Find equilibrium to the autonomous equations and classify them; y' = (y + 1 )(y^2 - 4) y' = 2y - 7 P' = r(1 - {P \over k...
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Closed quantum systems far from thermal equilibrium can show universal dynamics near attractor solutions, known as non-thermal fixed points, generically in the form of scaling behaviour in space and time. A systematic classification and comprehensive understanding of such scaling solutions are tasks of ...
If this were not true, if the probability were not strongly varying enough to make the great majority sit near the bottom, we would have to remember that oxygen gas is a mixture of O2 and single oxygen atoms in a nontrivial ratio. We know that there are, in fact, very few single ...
The probability of a stable nontrivial equilibrium in model (14)–(16) conditional on each unspecified function for maximal errors\(\varepsilon _g=0.1\),\(\varepsilon _f=0.005\), as depicted in Fig.2. Dark blue regions lie outside of the set of valid functions,V. (A)\(P\left( {...
(CNT)1. According to CNT, the equilibrium crystalline phase is nucleated randomly in a homogeneous disordered liquid by density fluctuations. Once the size of nuclei exceeds the critical size, they grow. The key thermodynamic factors for crystal nucleation are the free energy gain upon ...