Dynamics of the QR-flow for upper Hessenberg real matrices

Figure 1.  Left: Typical form of a block $ Y_{j,j} $ of $ \omega(X_0) $. Right: Reordered Jordan normal form used in the proof of Theorem 4.1. In both plots, the blocks labelled by O$ k $, $ k = 1,3,\dots $ (resp. by E$ k $, $ k = 2,4,\dots $), have simple eigenvalues $ \lambda \in \mathrm{Spec}({X_0}) $ with the same real part and odd (resp. even) multiplicity $ \mathrm{mult}({\lambda}) \geq k $. In the right plot, Id refers to the identity matrix of the corresponding dimension

Figure 2.  We consider $ X_0 $ from Example 4.2. Left: We display $ \log((X(t))_{4,3}) $ as a function of $ t $. The straight lines correspond to $ y = 5-x $ and $ y = 6-x $. Right: We display $ \log((X(t))_{6,5}) $ as a function of $ \log(t) $. The straight lines correspond to $ y = -2x $ and $ y = 1.8-2x $

Figure 3.  Phase portrait of the system (21) for $ c = d = 2 $. The triangular equilibrium matrices $ A_x $ correspond to the line of fixed points $ y = 2 $. The matrix $ C $ corresponds to the fixed point at $ (1,1) $

Figure 4.  Left: Evolution of the coefficient $ x_{11} $ of the trajectory of $ X_{0,\epsilon} $ as a function of the integration time $ t $ for $ \epsilon = 10^{-2} $ (green), $ \epsilon = 0 $ (blue) and $ \epsilon = 10^{-2} $ (red). For $ \epsilon = 10^{-2} $, $ x_{11} $ tends to be $ \pi $-periodic, reaching a sequence of maxima close to 8. For $ \epsilon = 0 $, $ x_{11} $ tends to be $ 2\pi $-periodic. For $ \epsilon>0 $, $ x_{11} $ tends to 2. Right: Projection onto the coordinates $ (x_{2,1},x_{2,2},x_{3,2}) $ of the $ \omega $-limit for $ X_{0,10^{-2}} $ (red), $ X_{0,-10^{-2}} $ (green), and the periodic orbit of $ X_{0,0} $ (blue). The $ \pi $-periodic orbit obtained as $ \omega $-limit for $ \epsilon> 0 $ (resp. for $ \epsilon <0 $) are observed in the coordinates plane $ x_{2,1} = 0 $ (resp. $ x_{3,2} = 0 $). For $ \epsilon = 0 $ the $ \omega $-limit is the $ 2\pi $-periodic orbit of $ X_{0,0} $

Figure 5.  From left to right, we show the orbit of $ X_0 $ in cases (ⅰ) $ \alpha = \pi/2,\ \beta = \pi/4 $, (ⅱ) $ \alpha = \pi/2, \ \beta = \pi/2 $ and (ⅲ) $ \alpha = \pi/4 $, $ \beta \approx 0.361367 $. In the first case the orbit is dens in a 2D torus. In the cases (ⅱ) and (ⅲ) the orbit is a periodic orbit

Figure 6.  We display the evolution of $ x_{2,1} $ with respect the time integration (top) and the projection onto the $ (x_{1,1},x_{2,1}) $-plane. Left: Orbit of $ X_0^{\mathbb{Q}} $. Right: Orbit of $ X_0^{\mathbb{R}\setminus \mathbb{Q}} $

Figure 7.  Homoclinic and heteroclinic orbits to $ X_{\pm,\pm} $. We represent some components of the solution $ X(t) $ of the QR-flow starting at $ X_i $, $ 1 \leq i \leq 8 $. In both plots we display the coefficient $ x_{2,3} $ in the $ y $-axis. Left: In the $ x $-axis we display the coefficient $ x_{1,1} $. The four equilibrium matrices are projected onto the points $ (0,1) $ and $ (0,-1) $. The homoclinic orbits appear as ellipses tangent to the origin. Each curve corresponds to two homoclinic or heteroclinic orbits (they are superimposed in this projection). Right: In the $ x $-axis we display the coefficient $ x_{1,2} $. The four equilibrium matrices are at the vertices of the square. The four homoclinic orbits start in one of the vertices, go to the center of the square and return. The four heteroclinic orbits are clearly observed

Figure 8.  Homoclinic and heteroclinic orbits to $ X_{i,j,k}^{+,+} $ (labelled by $ ijk $ in the top plots). In the bottom plots the points represent projections of more than one equilibrium matrix. Top left: Heteroclinic orbits to these equilibria. The two pieces of orbits which go to the bottom left vertex of the window correspond to the same heteroclinic orbit from $ X_{321}^{+,+} $ to $ X_{123}^{+,+} $. Top right: a scheme of the local homo/heteroclinic structure. Bottom left: The same of top plot, but in a larger window, to see the full heteroclinic orbit from $ X_{321}^{+,+} $ to $ X_{123}^{+,+} $. Bottom right: In the displayed projection each pair of points on the same horizontal coordinate in the top left plot project onto the same point