Long-time behaviour of the correlated random walk system

Figure 1.  Some eigenfunctions for $ S = 0.8 $, real and imaginary parts respect to $ x\in[-1/2, 1/2] $. From left to right, and up to down, plots from $ u_0, v_0 $, and real and imaginary part of $ u_{n, 1}, v_{n, 1} $ for $ n = 2, 4 $ (symmetric case, top), and $ n = 1, 3 $ (antisymmetric case, bottom). In all cases, the blue-cross line corresponds to $ u_{n, 1} $, and the red-circle one to $ v_{n, 1} $. The plots for the corresponding $ (u_{n, 2}, v_{n, 2}) $ are similar, so we do not include them. In these graphs, we can also see the oscillatory behaviour of the $ n $-th eigenfunction, described in Proposition 2.10 below

Figure 2.  $ \nu_{n, j}(S) $ (up) and $ \lambda_{n, j}(S) $ (bottom) in the complex plane for $ j = 1, 2, \ n = 0, 1, 2, 3, 4 $ (from left to right in the top row, and from the real axis up (or down), in the bottom row). We have $ S\in(0, 2] $ (first column) and $ S\in(0, 0.5] $ (second column). Note that the eigenvalues overlap on the real line. Also, as $ S $ increases, the values of $ \nu_{n, j}(S) $ and $ \lambda_{n, j}(S) $ become non-real, and $ \lambda_{n, j}(S) $ tend to $ -\infty\pm i\infty $

Figure 3.  $ {{ \rm{Re}}}(\lambda_{n, j}(S)) $ (top) and $ {{ \rm{Im}}}(\lambda_{n, j}(S)) $ (bottom) for $ j = 1, 2, \ n = 0, 1, 2, 3, 4 $ (top to down), with $ S\in(0, 2] $ (first column) and $ S\in(0, 0.5] $ (second column). Again, increasing $ S $ makes the values of $ \lambda_{n, j}(S) $, $ n\geq 1 $, become non-real and $ \lambda_0(S) $ and $ {{ \rm{Re}}}(\lambda_{n, j}(S)) $, $ n\geq 1 $, decrease to $ -\infty $. In this sense, we say that $ S $ is, indeed, a dissipative parameter: the larger it is, the faster the solutions tend to 0

Figure 4.  Intersections of the graphs of $ \sin x $, and $ Sx $ and $ -Sx $ for different values of $ S>0 $

Figure 5.  Graph of the function $ y(x) $ for $ S = 0.05 $. We observe that it is well defined and has a minimum in each $ \left(n\pi, (n+1)\pi\right) $ from $ n\geq 6 $ (which corresponds to $ \nu_{n, j}\notin\mathbb{R} $). The blue continuous line corresponds to the function obtained using (12), and the red discontinuous one corresponds to (13)

Figure 6.  From left to right and from top to bottom, eigenfunctions $ (u_{n, 1}, v_{n, 1}) $, $ n = 1, 2, 3, 4 $ for $ S = 0.8 $ in the complex plane. In all cases, the blue cross line corresponds to $ u_{n, 1} $, and the red circle one to $ v_{n, 1} $ We can see that the $ n $-th eigenfunction makes $ n $ complete half-turns. The plots for the corresponding $ (u_{n, 2}, v_{n, 2}) $ are similar