Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

Figure 1.  Example of Duffing potentials corresponding to different combinations of signs and different values of $ \kappa $. The $ (++) $ combination is referred to as hard Duffing potential while the combination $ (+-) $ is referred to as soft Duffing potential. (A combination $ (-+) $, not shown in this figure, is referred to as double-well potential.)

Figure 2.  Example of Keplerian potential with critical distance $ d_\mathrm{c} = 1 $

Figure 3.  Behavior of the simple pendulum (46) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.2\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.001 $

Figure 4.  Behavior of the simple pendulum (46) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.8\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

Figure 5.  Behavior of the hard Duffing oscillator (48) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.4\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0005 $

Figure 6.  Behavior of the hard Duffing oscillator (48) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.001 $

Figure 7.  Behavior of the soft Duffing oscillator (49) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0001 $

Figure 8.  Behavior of the soft Duffing oscillator (49) in the presence of non-linear damping, implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.2 $, $ \epsilon = 1.3 $ and $ h = 0.0001 $

Figure 9.  Behavior of the soft Duffing oscillator (50) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Heun method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0001 $

Figure 10.  Behavior of the soft Duffing oscillator (51) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Runge-Kutta method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ h = 0.0001 $

Figure 11.  Behavior of the double-well Duffing oscillator (52) in the absence of non-linear damping (namely, $ \mu = 0 $), implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.8 $ and $ h = 0.0001 $

Figure 12.  Behavior of the double-well Duffing oscillator (52) in the presence of non-linear damping, implemented through the Euler method. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [-1\ -1.5\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 1.4 $, $ \mu = 0.2 $, $ \epsilon = 1.3 $ and $ h = 0.0001 $

Figure 13.  Behavior of the Van der Pol oscillator (53) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.2\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.001 $

Figure 14.  Behavior of Van der Pol oscillator (53) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

Figure 15.  Behavior of the Kepler oscillator (54) in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0\ -0.3\ 0]^\top $. The values of the parameters used in this simulation are $ \lambda = 0.5 $, $ \rho = 0.08 $ and $ h = 0.001 $ (hence, the critical distance is $ d_\mathrm{c} = 0.4 $)

Figure 16.  Behavior of the Kepler oscillator (54) in the presence of non-linear damping. The left-hand panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point is $ x_0 = [0\ 0\ 1]^\top $, the reference point for the oscillator is $ r = [1\ 0\ 0]^\top $ (denoted by a blue open circle) and the initial speed $ v_0 = [0.5\ -0.9\ 0]^\top $. The values of the parameters used in this simulation are $ \rho = 0.08 $, $ \mu = 0.1 $, $ \epsilon = 2 $ and $ \lambda = 0.5 $ and $ h = 0.002 $ (hence, the critical distance is $ d_\mathrm{c} = 0.4 $)

Figure 17.  Behavior of the classic pendulum oscillator (58) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.002 $

Figure 18.  Behavior of the simple pendulum (58) in the presence of non-linear damping. The left-hand panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.3 $, $ \epsilon = 1.3 $ and $ h = 0.002 $

Figure 19.  Behavior of a hard Duffing oscillator (59) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $ and $ h = 0.0005 $

Figure 20.  Behavior of a hard Duffing oscillator (59) in the presence of non-linear damping. The left-hand panel shows the trajectory in the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the refernce point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 0.5 $, $ \mu = 0.5 $, $ \epsilon = 1.3 $ and $ h = 0.0008 $

Figure 21.  Behavior of the Van der Pol oscillator (60) on the special orthogonal group $ SO(3) $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the initial point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 1 $ and $ h = 0.0005 $

Figure 22.  Behavior of the Van der Pol (60) in the presence of non-linear damping. The left-hand panel shows the trajectory in the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the initial point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \kappa = 1 $, $ \mu = 0.5 $, $ \epsilon = 3 $ and $ h = 0.001 $

Figure 23.  Behavior of the Kepler oscillator (61) on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, in the absence of non-linear damping (namely, $ \mu = 0 $). The left-hand side panel shows the trajectory in the space $ {{{\mathbb{S}}}}^{2} $, when the starting point and also the reference point are taken randomly, (denoted by a blue open circle) and the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \lambda = 0.2 $, $ \rho = 0.8 $ and $ h = 0.0003 $

Figure 24.  Behavior of the Kepler oscillator (61) in the presence of non-linear damping. The left-hand panel shows the trajectory on the special orthogonal group $ {{{\mathrm{SO}(3)}}} $, when the starting point and also the reference point are taken randomly (denoted by a blue open circle) the initial speed will be random, because related to the initial state. The values of the parameters used in this simulation are $ \lambda = 0.5 $, $ \mu = 0.3 $, $ \rho = 0.1 $, $ \epsilon = 1.5 $ and $ h = 0.0003 $

Figure 25.  Behavior of the classic pendulum oscillator (31) on two different manifolds, in the absence of non-linear damping (namely, $ \mu = 0 $). Indicating by $ k $ the discrete-time index, the figure (a) shows the configuration of the system at $ k = 2,301 $, the figure (b) at $ k = 4,461 $, the figure (c) at $ k = 9,321 $ and the figure (d) shows where the simulations stops, at $ k = 15,000 $

Figure 26.  Behavior of the classic pendulum oscillator (31) on two different manifolds, in the presence of non-linear damping. Indicating by $ k $ the discrete-time stamp, the figure (a) shows the configuration of the system at $ k = 2,641 $, the figure (b) at $ k = 6,381 $, the figure (c) at $ k = 10,381 $ and the figure (d) shows where the simulations stops, at $ k = 15,000 $