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# Computing complete Lyapunov functions for discrete-time dynamical systems

The research in this paper was supported by the Icelandic Research Fund (Rannís) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation

• A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction.

We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.

Mathematics Subject Classification: Primary: 93D30, 65D12, 65K10; Secondary: 39A30.

 Citation: • • Figure 1.  Example (24) with solving $\Delta v(x,y) = -1$. Chain-recurrent set (top) approximated by the set $\{(x,y)\, \mid\, \Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (middle) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$. $\Delta v$ is approximately zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: Constructed complete Lyapunov function $v(x,y)$, which has a minimum at the origin

Figure 2.  Example (24) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ (middle). Again, the orbital derivative $\Delta v$ is correctly approximated being zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: Constructed complete Lyapunov function $v(x,y)$, which has a minimum at the origin. The point with equality constraint was $(0.5,0)$

Figure 3.  Example (24) with inequality constraints. Chain recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ (middle). $\Delta v$ is approximately zero on the chain-recurrent set (origin) and negative everywhere else. Bottom: The constructed complete Lyapunov function $v(x,y)$, which has a minimum at the origin

Figure 4.  Example (25) with solving $\Delta v(x,y) = -1$. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The approximated chain-recurrent set includes the equilibria at the origin and $(\pm 1,0)$, but is much larger, in particular around the origin. Bottom: Constructed complete Lyapunov function $v(x,y)$, which has a saddle point at the origin and a local maximum at the unstable equilibria $(\pm 1,0)$

Figure 5.  Example (25) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. $\Delta v$ is approximately zero on the chain-recurrent set, consisting of three equilibria at the origin and $(\pm 1,0)$, and negative everywhere else. The approximation of the chain-recurrent set (the equilibria) is much better than when solving the equation $\Delta v(x,y) = -1$. Bottom: Constructed complete Lyapunov function $v(x,y)$, which has a saddle point at the origin and local maxima at the unstable equilibria $(\pm 1,0)$. Note that they have different levels, which is due to the extra point with equality constraint at $(0.5,0)$, resulting in an unsymmetric approximation

Figure 6.  Example (25) with inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. $\Delta v$ is approximately zero on the chain-recurrent set, consisting of three equilibria at the origin and $(\pm 1,0)$, and negative everywhere else. The approximation of the chain-recurrent set (the equilibria) is much better than when solving the equation $\Delta v(x,y) = -1$. Bottom: Constructed complete Lyapunov function $v(x,y)$, which has a saddle point at the origin and local maxima at the unstable equilibria $(\pm 1,0)$

Figure 7.  Example (26) with solving $\Delta v(x,y) = -1$. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$. The approximated chain-recurrent set does not resemble the Hénon attractor very well, neither using the orbital derivative nor as the local minimum of the constructed function

Figure 8.  Example (26) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The characteristic shape of the Hénon attractor is clearly visible. Bottom: Constructed complete Lyapunov function $v(x,y)$ with a local minimum at the Hénon attractor. The point with equality constraint was $(0.5,0)$

Figure 9.  Example (26) with inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. The characteristic shape of the Hénon attractor is clearly visible. Bottom: Constructed complete Lyapunov function $v(x,y)$ with a local minimum at the Hénon attractor

Figure 10.  Example (27) with solving $\Delta v(x,y) = -1$. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$. The approximated chain-recurrent set shows the Hénon repeller better than the Hénon attractor in the previous example, but still not very clearly. It is not clearly visible as local maximum of the constructed function either

Figure 11.  Example (27) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$, showing the Hénon repeller as a local maximum. The repeller is clearly visible in all figures. The point with equality constraint was $(0.5,0)$

Figure 12.  Example (27) with inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$, showing the Hénon repeller as a local maximum. The repeller is clearly visible in all figures

Figure 13.  Example (28) with solving $\Delta v(x,y) = -1$. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$. The approximated chain-recurrent set shows the attractor relatively well in the orbital derivative, but not very clearly as local minimum of the constructed function

Figure 14.  Example (28) with equality and inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$, showing the attractor as a local minimum. The attractor is clearer than in the previous method, both using the orbital derivative and as local minimum of the constructed function. The point with equality constraint was $(0.5,0)$, where the orbital derivative is fixed at $-1$

Figure 15.  Example (28) with inequality constraints. Chain-recurrent set (top) approximated by the set $\{(x,y)\,\mid\,\Delta v(x,y)\ge \gamma\}$, see Table 2, and the orbital derivative (second) $\Delta v(x,y)$ of the constructed complete Lyapunov function $v$ over the chain-recurrent set. The third figure shows the orbital derivative in a larger set. Bottom: Constructed complete Lyapunov function $v(x,y)$, showing the attractor as a local minimum. The attractor is clearer than in the first method, both using the orbital derivative and as local minimum of the constructed function

Figure 16.  Example (29) with solving $\Delta v(x,y,z) = -1$. Top: Chain-recurrent set approximated by the set $\{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\}$, see Table 2. The other figures show projections of this set: projections to the $xy-$ (second), $yz-$ (third) and $xz-$plane (bottom)

Figure 17.  Example (29) with equality-inequality constrains. Top: Chain-recurrent set approximated by the set $\{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\}$, see Table 2. The other figures show projections of this set: projections to the $xy-$ (second), $yz-$ (third) and $xz-$plane (bottom). The figures are not as good as with the previous method. The point with equality constraint is $(0.4,0.4,0)$

Figure 18.  Example (29) with inequality constrains. Top: Chain-recurrent set approximated by the set $\{(x,y,z)\,\mid\,\Delta v(x,y,z)\ge \gamma\}$, see Table 2. The other figures show projections of this set: projections to the $xy-$ (second), $yz-$ (third) and $xz-$plane (bottom)

Table 1.  Collocation points $X$ for the examples. We have used $N$ collocation points in a hexagonal grid with parameter $\alpha_{\text{Hexa-basis}}$ within a rectangle $(x,y)\in [x_{\rm min},x_{\rm max}]\times [y_{\rm min},y_{\rm max}]$ or $(x,y,z)\in [x_{\rm min},x_{\rm max}]\times [y_{\rm min},y_{\rm max}]\times [z_{\rm min},z_{\rm max}]$ for the three-dimensional example (29). The number of evaluation points is also displayed

 Example $N$ $\alpha_{\text{Hexa-basis}}$ #-evaluation $x_{\rm min}$ $x_{\rm max}$ $y_{\rm min}$ $y_{\rm max}$ $z_{\rm min}$ $z_{\rm max}$ (24) $3,584$ $0.072$ 1,779,556 $-2$ $2$ $-2$ $2$ (25) $10,108$ $0.03$ 2,003,001 $-2$ $2$ $-1$ $1$ (26) $1,440$ $0.05$ 1,334,000 $-1.5$ $1.5$ $-0.5$ $0.5$ (27) $5,520$ $0.025$ 1,334,000 $-1.5$ $1.5$ $-0.5$ $0.5$ (28) $2,900$ $0.08$ 1,779,556 $-2$ $2$ $-2$ $2$ (29) $5,301$ $0.07$ 1,030,301 $-0.2$ $0.9$ $-0.2$ $0.9$ $-0.2$ $0.9$

Table 2.  The value of the parameter $\gamma\le 0$, close to $0$, for all examples. The chain-recurrent set is approximated by the set $\left(\Delta v \right)^{-1}([\gamma,\infty))$

 $\gamma$ for each method System $\Delta v(x)=-1$ equality-inequality inequality (24) $-0.1$ $-10^{-5}$ $0$ (25) $-0.2$ $-10^{-5}$ $-10^{-4}$ (26) $-0.1$ $0$ $-10^{-2}$ (27) $-0.1$ $0$ $0$ (28) $-0.1$ $0$ $0$ (29) $-0.1$ $0$ $0$
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