Fixed points linear stability

A fixed point of a map is linearly stable if and only if all eigenvalues of the Jacobian satisfy A <1. Determine the stability of the fixed points of the Henon map, as a function of a and b. Show that one fixed point is always unstable, while the other is stable for a slightly larger than Show that this fixed point loses stability in a flip bifurcation (A = -1) at a, = (1 - b. ... 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

In the two-dimensional case (two variables) "almost any C1-smooth dynamic system is rough (i.e. at small bifurcations its phase pattern deforms only slightly without qualitative variations). For rough two-dimensional systems, the co-limit set of every motion is either a fixed point or a limit cycle. The stability of these points and cycles can be checked even by a linear approximation. Mutual relationships between six different types of slow relaxations for rough two-dimensional systems are sharply simplified. 

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

So far we have relied on graphical methods to determine the stability of fixed points. Frequently one would like to have a more quantitative measure of stability, such as the rate of decay to a stable fixed point. This sort of information may be obtained by linearizing about a fixed point, as we now explain. 

Using linear stability analysis, determine the stability of the fixed points for X =sinx. 

Classify the fixed points of the logistic equation, using linear stability analysis, and find the characteristic time scale in each case. 

Use linear stability analysis to classify the fixed points of the following systems. If linear stability analysis fails because f x ) = 0, use a graphical argument to decide the stability. 

Using linear stability analysis, classify the fixed points of the Gompertz model of tumor growth TV = -aN n(bN). (As in Exercise 2.3.3, TV(r) is proportional to the number of cells in the tumor and a,b>Q are parameters.)... 

Give a linear stability analysis of the fixed points in Figure 3.1.5. 

We now see that a supercritical pitchfork bifurcation occurs at 7 = 1. It s left to you to check the stability of the fixed points, using linear stability analysis or graphical methods (Exercise 3.5.2). 

Do the linear stability analysis for all the fixed points for Equation (3.5.7), and confirm that Figure 3.5.6 is correct. 

We ve already seen a simple instance of hyperbolicity in the context of vector fields on the line. In Section 2.4 we saw that the stability of a fixed point was accurately predicted by the linearization, as long as f x ) 0. This condition is the... 

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. 

Consider the map x ., =sinx . Show that the stability of the fixed point x = 0 is not determined by the linearization. Then use a cobweb to show that x = 0 is stable—in fact, globally stable. 

Conveniently, an investigation of the dynamic behavior of a set of differential equations starts out with the determination of the fixed points and their stability. The latter is studied hy linearizing the system s equation about the steady state and then evaluating the temporal evolution of small perturbations. Denoting the perturbations by 8( )ql and 8c, in our case the equations read ... 

The last panel in Fig. 3.4 shows a situation qualitatively different from the previous ones. Here, the FN nullclines intersect at three points. Simple linear stability analysis reveals that the central one is unstable, whereas the lateral ones are both stable. Since there are no more attracting structures in this simple dynamical system, the final state of the FN dynamics will be one of these stable fixed points, depending on the initial condition. This situation is called bistability. 

To achieve a deeper understanding of the stability properties of the inhomogeneous fixed point under the influence of the control force and to obtain the general form of the characteristic equation which determines the eigenvalues of this linearized system, we perform the linearization of the original continuous system (5.30) at the spatially inhomogeneous fixed point (ao(x),uo). Introducing... 

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

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>

The asymptotic stability of a fixed point can be determined by using the linearization approach. By adding a distortion y, to the fixed point x f and definition of... 

A local stability analysis can be performed on the transformation defined in Eq. (31). In the neighborhood of the fixed point the linear stability matrix is... 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

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