Analysis near critical points or cycles

Linear systems by themselves have a rather simple dynamical behavior. The reason, why linear system analysis is so important, is that non-linear systems are often analyzed by local linearization [83], i.e., a non-linear function is approximated (locally) by a linear function. Linearization is typically done by using a Taylor expansion of a function and neglecting the higher-order terms. See Eq. 3.14 for Taylor expansion of a scalar function $f:\mathbf{R}\to\mathbf{R}$, Eq. 3.15 shows the Taylor expansion for a vector function $\mathbf{f}:\mathbf{R}^n\to\mathbf{R}^n$. This kind of analysis is especially easy near critical points, since the long-term behavior trivially coincides with the local behavior at these points.

  

$$\sum_{i\ge0}\ \frac{1}{i!} \ d^i\cdot{}f^{(i)}(c) \approx f(c) + d\cdot{}f'(c)$$ f(c+d) = (3.13)

$$\sum_{i\ge0}\ \frac{1}{i!} \left.(\mathbf{d}\cdot\nabla)^{i}\,\ma... ...f{f}(\mathbf{c}) + \mathbf{d}\cdot\left.\nabla\mathbf{f}\right\vert _\mathbf{c}$$ f(c+d) = (3.14)

Dynamical system analysis near a critical point

Analyzing the system's behavior near its critical points can help to understand the evolution of any state of the system. Assuming the system is non-linear and hyperbolic, linearization can be used to determine the behavior near critical points completely. Continuous and discrete systems can be treated rather similar [72] (See Tab. 3.2).

 

Table 3.2: Local linearization near critical points

	continuous case	discrete case
vector field def.	$\dot{\mathbf{x}} = \mathrm{d}\mathbf{x}/\mathrm{d}t = \mathbf{f}(\mathbf{x})$	$\Delta\mathbf{x}_n = \mathbf{f}(\mathbf{x}_n)$	(a)
critical point def.	$\dot{\mathbf{c}} = \mathbf{f}(\mathbf{c}) = 0$	$\Delta\mathbf{c} = \mathbf{f}(\mathbf{c}) = 0$	(b)
re-writing x, xn	x = c+d	xn = c+dn	(c)
using Taylor exp.	$\dot{\mathbf{x}} \approx \left.\nabla\mathbf{f}\right\vert _\mathbf{c}\cdot\mathbf{d}$	$\Delta\mathbf{x}_n \approx \left.\nabla\mathbf{f}\right\vert _\mathbf{c}\cdot\mathbf{d}_n$	(d)
linearized system	$\dot{\mathbf{d}} = \left.\nabla\mathbf{f}\right\vert _\mathbf{c}\cdot\mathbf{d}$	$\Delta\mathbf{d}_n = \left.\nabla\mathbf{f}\right\vert _\mathbf{c}\cdot\mathbf{d}_n$	(e)

 

To keep the analysis simple, we assume the system to be autonomous and time-independent (see Tab. 3.2(a) for the definitions). Assuming the existence of at least one critical point (see Tab. 3.2(b) for the definitions) any state of the dynamical system near critical point c can be rewritten with respect to c (see Tab. 3.2(c)). With this reformulation the dynamical system can be approximated by a Taylor expansion as shown in Tab. 3.2(d). $\left.\nabla\mathbf{f}\right\vert _\mathbf{c}$ denotes the Jacobian matrix of f(x) evaluated at c. Using Tab. 3.2(c) again, the left side of the Taylor expansion in Tab. 3.2(d) can be rewritten. This operation yields the linearized systems for small perturbations around critical point c (see Tab. 3.2(e)). These linear systems can now be analyzed as discussed in the last section.

Dynamical system analysis near a cycle

Cycles are another important class of characteristic subsets within continuous dynamical systems. A cycle is given, when the system returns to a previous state. The system behavior near such a cycle can be analyzed by using a Poincaré map. Such a map is a discrete dynamical system, that is produced from a continuous dynamical system and that is of a lower dimension than the original system. A Poincaré map is specified by the cross-section of a surface perpendicular to the cycle (usually a plane) and a trajectory near the cycle. The Poincaré map is a discrete dynamical system with at least one critical point c, i.e., c is the intersection of the cycle and the surface. Thus the Poincaré map can be analyzed as shown in the section before and the results are then used for interpreting the system's behavior nearby the cycle [72].

mailto:helwig@cg.tuwien.ac.at.