Oscillations singular point

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

When pK > 4(32 holds, the singular point remains stable, Reei, 2 < 0, but the roots (2.1.16) have imaginary parts Imei = Im ei. In this case the phase portrait reveals a stable focus - Fig. 2.2. This regime results in damped oscillations around the equilibrium point (2.1.24). The damping parameter pK/(3 is small, for large 3, in which case the concentration oscillation frequency is just ui = y/pK. ... 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

Mathematical models of the reaction yield various solutions. Some of the solutions obtained are One singular point, 3 singular points, oscillating limit cycle, double periodic oscillations, chaotic oscillations. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

One singular point, two singular points, three singular points and their stability, as well as stable periodic solutions (sustained oscillations). 

The zero-point oscillations of the energy density of plane waves of photons have the same magnitude everywhere. In contrast, those calculated in the presence of a singular point (source or absorber) manifest spatial inhomogeneity. Precisely, the vacuum noise is concentrated in some vicinity of the singular point. 

In contrast to (143), this is a diagonal matrix independent of the spatial variables. Hence, in exactly the same way as with the zero-point oscillations of energy density, the vacuum fluctuations of polarization in empty space has the global nature, while those in the presence of the singular point manifest certain spatial inhomogeneity. 

A perturbation analysis of Equations 35 and 36 about this singular point shows that the solutions whose initial conditions are close to P, Z, oscillate sinusoidally about this singular point. Hence, no constant solution is possible. The prey and predator populations continually oscillate and are out of phase with each other. When the predator predominates, the prey is reduced, which in turn causes the predator to die for lack of food, which allows the prey to proliferate for lack of predator, which then causes the predator to grow because of the prey available as a food supply, and so on. The interesting feature is that these oscillations continue indefinitely. 

Since for P0 > 0, these roots have negative real parts, this singular point is a stable focus, and the steady state values given by Equation 40 are approached either by a damped sinusoid or an exponential (63). Note that for P0 — 0, the classical case, the roots are purely imaginary, and the oscillation persists indefinitely. 

Experiments carried out on mitosis in Physarum polycephalum (Kaufmann Wille, 1975 Tyson Sachsenmaier, 1978) did not allow a definitive conclusion in favour of one of the two views. The difficulty came from the fact that, short of the identification of the true biochemical variables driving the cell cycle, it was difficult to demonstrate the existence of a limit cycle in Physarum. Experiments based on the fusion of two plasmodia taken at different phases of the mitotic cycle or on the effect of heat shocks aimed at demonstrating the existence of a singular point from which the limit cycle would be reached with an indefinite phase (Winfree, 1974,1980,1987). These experiments did not allow the distinction to be made between a limit cycle characterized by relaxation oscillations and a discontinuous mechanism of the type discussed above. 

This is the heat capacity of a one-dimensional oscillator according to Einstein. The heat capacity deviates at low temperatures. It is not possible to expand into a Taylor series around T 0. In other words, the function has a pole at zero, which emerges as an essential singular point. A more accurate formula is due to Debye, n... 

A positive value of the Jacobian (1.19) in the bifurcation point is one of the conditions for oscillatory regime. This is an equivalent of the assertion that the self-oscillations are most probable in the systems where crossed feedbacks have opposite signs. There are some other methods to identify the self-oscillatory systems direct application of Hopf theorem, analysis of type of singular points, Bendixson criterion, reduction of the equation system to Lienard equation, and others. One can find details, for example, in Chap. 4 of [53], or elsewhere [65]. 

The decay constants are imaginary The displacement variables, being linear combinations of the functions and e (or equivalently sin (ot and cos cot), do not describe relaxation to a steady state. Instead a system perturbed from the steady state undergoes perpetual oscillation. The origin in the i-rj plane (i.e., X = Xq,Y = Fq) is not a steady state at all rather it is a singular point which is not accessible. Even if the system were prepared at the steady state, it could not remain there. Due to the fluctuations which constantly arise, a system in which initially = rj = 0 is unstable. Fluctuations create displacements and oscillations commence. 

The condition (i) implies that the modulus VFi vanishes at the centre of the spiral. Thus the centre of the spiral is a singular point of the structure as the phase of oscillations cannot be defined at this point. The latter is called a topological defect and possesses a topological charge equal to m. In the sequel, we consider only spirals for which m = lorm = -l, i.e., one-arm spirals, respectively left-handed or right-handed. 

The corresponding quantum mechanical expression of s op in Equation (4.19) is similar except for the quantity Nj, which is replaced by Nfj. However, the physical meaning of some terms are quite different coj represents the frequency corresponding to a transition between two electronic states of the atom separated by an energy Ticoj, and fj is a dimensionless quantity (called the oscillator strength and formally defined in the next chapter, in Section 5.3) related to the quantum probability for this transition, satisfying Jfj fj = l- At this point, it is important to mention that the multiple resonant frequencies coj could be related to multiple valence band to conduction band singularities (transitions), or to transitions due to optical centers. This model does not differentiate between these possible processes it only relates the multiple resonances to different resonance frequencies. 

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