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Centres and foci
Next special case we consider is the dynamical system of the form
Fig. 8.9. Phase portrait for dynamical system (8.20). Blue line represents unstable manifold, red line represents stable manifold.
Matrix of this system is
System (8.21) is little trickier to solve. Let us switch to polar coordinate system by usual transformation
x = r cos θ; y = r sin θ;
where r = r(t) and θ = θ(t). Inverse transformation reads
These relations can be used to find
Now we use (8.21) to derive corresponding equations for r and θ:
We can see that dynamical system (8.21) in polar coordinates decouples to two independent equations for coordinates r and θ,
First we solve equation for r. Let us write it in the form
which integrates to
log r = α t + log C
where the integration constant has been written as a logarithm (see footnote on page 162). Exponentiating the last equation we arrive at
Obviously, at time t = 0 we have r(0) = C and so we write the solution in the form
Next we solve equation for θ. This is trivial since we have
which integrates to
where the integration constant has been denoted by θ0 and represents the value of θ at t = 0. Summa summarum, solution of system (8.23) acquires the form
Hence, solution of original system (8.21) in the Cartesian coordinates reads
Suppose that α = 0 so that
Clearly, this represents motion at constant angular velocity β and constant radius r0 and therefore the phase trajectories are circles of radius r0. If
α ≠ 0, the radius of the "circle" will be
and hence the trajectory will be a spiral. If α > 0, the radius will increase exponentially and the spiral will tend to in nity. If, on the other hand, α < 0, the radius will decrease exponentially and the phase tra jectories will spiral towards the origin. All cases are plotted in gure 8.10 by Mathematica commands
and can be classi ed as follows:
Parameter β has the meaning of angular velocity. If it is zero, spirals become straight lines and dynamical system reduces to previous case (8.17). If it is non-zero, its sign determines the sense of rotation: tra jectories orbit the origin in a clockwise sense for β > 0 and in a counter-clockwise sense for β < 0.
Let us now analyse critical points of system (8.21) in terms of eigenvalues of matrix (8.22)
We can use Mathematica to nd the eigenvalues and eigenvectors of matrix (8.22) by
which shows that this matrix has two eigenvalues
In other words, eigenvalues and eigenvectors of matrix J satisfy relations
The rst observation is that the eigenvectors are complex and hence there are no neither stable nor unstable manifolds, i.e. there is no real direction which is mapped to the same direction. The only exception is when β = 0 since in this case dynamical system (8.21) reduces to (8.17) and the eigenvectors become real.
Second, eigenvalues λ1,2 are mutually complex conjugated (as well as the eigenvectors),
Fig. 8.10. Classi cation of critical points for the system (8.21): a, b) centre, c) unstable focus, d) stable focus.
where the bar denotes the complex conjugation. Hence, even if the dynamical system is not of the form (8.21), we can conclude, that if the matrix J has two complex conjugated eigenvalues
the critical point is stable/unstable focus or a centre, depending on the values of α and β as classi ed above.
Example. Consider dynamical system
This system is not of the form (8.21) but we can apply the criterion based on the analysis of eigenvalues. In Mathematica we type
where we have used Expand in order to simplify the expression for eigenvectors (try this code without Expand). We have found two eigenvalues
which are mutually complex conjugated. In this case, parameters α and β are
Parameter α is positive and so the critical point is an unstable focus. Tra jectories of dynamical system considered:
Another example is the system
Eigenvalues are found by
Hence, now the eigenvalues are
which means that
Since α = 0, critical point is a centre rather than focus. Tra jectories of this dynamical system are the following:
In the previous two sections we studied two special cases of planar linear dynamical systems given by matrices
However, we have seen that the analysis can be performed using the eigenvalues of these matrices. Now we consider general linear planar dynamical system
Let us find the eigenvalues and eigenvectors of this general matrix. Recall that the determinant of matrix J is
The trace of the matrix is de ned as a sum of its diagonal elements, i.e.
Eigenvalues λ are defined by equation
where e is an eigenvector. The last equation can be rewritten in the form
where I is the unit matrix 2 x 2 so that
This equation is a homogeneous system of linear equations which has non-trivial solutions only if the determinant of the system is zero:
This determinant reads
Expanding the brackets we arrive at
This is a quadratic equation for λ and its solutions are
Now we can summarize the classification of critical points as follows.
Moreover, if the real parts of eigenvalues λ1,2 are non-zero, critical point is called hyperbolic, otherwise it is called non-hyperbolic.