Article Contents
Article Contents

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

• * Corresponding author: Simone Fiori
• The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.

Mathematics Subject Classification: 37J15, 37M05, 37N35.

 Citation:

• 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 $k = 15,000$

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