Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

Figure 1.  Attractors of Yang-Chen system with (a) $ a = 10 $, $ b = 8/3 $, and $ c = 16 $; (b) $ a = 35 $, $ b = 3 $, and $ c = 35 $

Figure 2.  Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $ U_1 $ for (blue) $ (a, b, c) = (0.5, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03) $, (red) $ (a, b, c) = (1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03) $, (black) $ (a, b, c) = (0.1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.01) $, respectively

Figure 3.  Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $ V_1 $ (blue) $ (a, b, c) = (0.5, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03) $, (red) $ (a, b, c) = (1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03) $, (black) $ (a, b, c) = (0.1, 1.01, 1) $ with initial conditions $ (z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.01) $, respectively

Figure 4.  Phase portrait of the system (10), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $ U_2 $

Figure 5.  Phase portrait of the system (12), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $ U_3 $

Figure 6.  Phase portrait of system (1) at infinity

Figure 7.  Time variation of the deviation vector and its curvature near $ E_{1} $, for $ a = 35 $, $ b = 3 $

Figure 8.  Time variation of instability exponent $ \delta(E_{1}) $ for $ a = 35 $, $ b = 3 $, and different values of $ c $

Figure 9.  Time variation of the deviation vector and its curvature near $ E_{2, 3} $ with $ a = 35, b = 3 $. Initial conditions used to integrate deviation equations are $ \xi_{1}(0) = \xi_{2}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = 10^{-6} $

Figure 10.  Time variation of curvature $ \kappa_{0} $ of deviation vector near equilibrium points $ E_{1} $ with $ a = 35, b = 3 $

Figure 11.  Time variation of curvature $ \kappa_{0} $ of deviation vector near equilibrium points $ E_{2, 3} $ with $ a = 35, b = 3 $

Figure 12.  A large version of Fig. 11 at time $ 0.25 $ to $ 0.55 $