Fixed point of order

Figure 13 Pbint dynamics on the Poincare map. (A) A typical mapping sequence of a point initially at (p2>

To analyze the stability properties of motion nearby a fixed point on the map, we will consider a linear approximation to the mapping dynamics near the fixed point. For simplicity we examine a fixed point of order 1. As before, let (p2, (Jz) l e fixed point let (p2> differential vector hpz, hqz), and let (p z, q z) be the first iterate of pz, qz) generated by initiating a trajectory at that point, as in Figure 15. 

Figure IS Numerical linear stability analysis of a fixed point of order 1. A point (P2>

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

Generalizing the second observation to cycles of arbitrary length, we note that since each primitive string in a limit cycle with least period equal to q must be a fixed point of the g order rule, (and thus also a fixed point of [4>Y, for any period p with q p), each primitive limit cycle may be expressed by a term equal to its period ... 

The above scenario is accounted for by the normal form (4.9) truncated at fourth order in q with k = v = a = p = 0 and x < 0, taking p as the bifurcation parameter, which increases with energy (p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8) are given by p = 0 and dv/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. 

X2 is attracting whereas is repelling. This accounts for the sharp corner in the graph of Xoo r) at r — tq. For r > ri = 3, neither of the two fixed points of fr is stable. In order to understand the period doubling at r = ri, it is necessary to consider the fixed points (and their stability properties) of the second iterate of fr. The second iterate f x) is a quartic polynomial in x. Therefore, the fixed point equation X xf ) has four solutions given by... 

Recall that in the previous section P, denoted the periodic solutions obtained in Proposition 3.2, whereas now we reserve this same notation for the corresponding fixed points of the Poincare map. In order to avoid confusion we will write ,(/) from now on whenever we wish to refer to the solutions. 

The second approach is the phenomenological renormalization (PR) [24,70] method, where the sequence of the pseudocritical values of X can be calculated by knowing the first and the second lowest eigenvalues of the matrix for two different orders, N and N1. The critical Xc can be obtained by searching for the fixed point of the phenomenological renormalization equation for a finite-size system [70],... 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

The Newtonian gravitational force is the dominant force in the N-Body systems in the universe, as for example in a planetary system, a planet with its satellites, or a multiple stellar system. The long term evolution of the system depends on the topology of its phase space and on the existence of ordered or chaotic regions. The topology of the phase space is determined by the position and the stability character of the periodic orbits of the system (the fixed points of the Poincare map on a surface of section). Islands of stable motion exist around the stable periodic orbits, chaotic motion appears at unstable periodic orbits. This makes clear the importance of the periodic orbits in the study of the dynamics of such systems. 

In order to calculate the ratio 5 between the semi-minor and the semi-major axes of the ellipses surrounding the fixed points of a periodic orbit we consider a two dimensional area preserving mapping and a point [x, y) of a periodic orbit of frequency P/Q. Let us write the Jacobian of the Qth iteration of the map as follows ... 

Hence, a solves the first-order conditions if and only if it is a fixed point of mapping f[x) defined above. 

The open-loop strategy implies that each players control is only a function of time, Ui = Ui t). A feedback strategy implies that each players control is also a function of state variables, ui = Ui t Xi t) Xj(t)). As in the static games, NE is obtained as a fixed point of the best response mapping by simultaneously solving a system of first-order optimality conditions for the players. Recall that to find the optimal control we first need to form a Hamiltonian. If we were to solve two individual non-competitive optimization problems, the Hamiltonians would be Hi = fi XiQi, i = 1,2, where Xi t) is an adjoint multiplier. However, with two players we also have to account for the state variable of the opponent so that the Hamiltonian becomes... 

The saddle equilibrium states are the saddle fixed points of the shift map, and respectively, their separatrices are the invariant manifolds. Returning to the original (non-rescaled) variables we find that the fixed points must lie apart from the origin at some distance of order e. If the third iteration (10.6.2) of the map (10.6.1) were the shift map of the reduced system (10.6.5), then the above theorem would follow from our arguments because the fixed points Oi, O2,03 of the third iterations correspond to the cycle of period three of the original map. 

In fact, no common upper bound exists on the number of the periodic orbits which can be generated from a fixed point of a smooth map through the given bifurcation. If the smoothness r of the map is finite, the absence of this upper estimate is obvious because it follows from the proof of the last theorem that to estimate the number of the periodic orbits within the resonant zone 1/ = M/N the map must be brought to the normal form containing terms up to order (AT — 1). In this case the smoothness of the map must not be less than (iV — 1). Hence, we can estimate only a finite number of resonant zones if the smoothness is finite. 

Suppose that, near some fixed point G F, dT, the graph F, is a straight line segment parallel to the x axis. Let G (0, T) be an arbitrary fixed point and let Re C denote the ball of a sufficiently small radius with centre (a °,t°). First, we examine the smoothness of the function X = (IF, w). Let D stand for a first-order derivative and let (p denote an arbitrary smooth function in i 2s such that p = 0 outside Rzeji 0 < (> < 1, and dpidy = 0 on F. ... 

Fig. 5.2 Plots of 11 iV -order N < 10) LST approxirnatioii.s of the invariant (ie., fixed point) density, p for elementary rule R22 see text.

In view of the problems referred to above in connection with direct potentiometry, much attention has been directed to the procedure of potentio-metric titration as an analytical method. As the name implies, it is a titrimetric procedure in which potentiometric measurements are carried out in order to fix the end point. In this procedure we are concerned with changes in electrode potential rather than in an accurate value for the electrode potential with a given solution, and under these circumstances the effect of the liquid junction potential may be ignored. In such a titration, the change in cell e.m.f. occurs most rapidly in the neighbourhood of the end point, and as will be explained later (Section 15.18), various methods can be used to ascertain the point at which the rate of potential change is at a maximum this is at the end point of the titration. 

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