Detection meeting control: Unstable steady states in high-dimensional nonlinear dynamical systems

We articulate an adaptive and reference-free framework based on the principle of random switching to detect and control unstable steady states in high-dimensional nonlinear dynamical systems, without requiring any a priori information about the system or about the target steady state. Starting from an arbitrary initial condition, a proper control signal finds the nearest unstable steady state adaptively and drives the system to it in finite time, regardless of the type of the steady state. We develop a mathematical analysis based on fast-slow manifold separation and Markov chain theory to validate the framework. Numerical demonstration of the control and detection principle using both classic chaotic systems and models of biological and physical significance is provided.

Figure 1

(a) A chaotic attractor of the Lorenz system with three USSs (highlighted). (b) Time series $x_2\left(t\right)$ of the controlled Lorenz system starting from different initial conditions. (c) Approximate distributions of the “basin” volumes of the USSs. The probability of convergence to each USS is calculated using 100 independent control realizations. For the previous method, the time series $x_2\left(t\right)$ is used: $\omega \left(t\right)\equiv {\omega }_{+}$ (d) and $\omega \left(t\right)\equiv {\omega }_{-}$ (e), where ${\omega }_{+}=1$, ${\omega }_{-}=-1$, $k=130$, and time window $\mathrm{\Delta }T=1$. The initial conditions are from the subspace defined by $x_1\left(0\right)=x_2\left(0\right)=x_3\left(0\right)=y_2\left(0\right)$.

Figure 2

For the controlled Lorenz system, an example of fast-slow manifold separation, where the dotted line is calculated by setting ${\dot{x}}_1={\dot{x}}_2={\dot{x}}_3=0$. Four different trajectories are shown.

Figure 3

Schematic illustration of random switching control. (a) Function $h\left(y\right)-y$ and two neighboring nondegenerated fixed points $y_1^{*}$ and $y_2^{*}$. (b,c) Dynamics of the system $\dot{y}=\omega \left(t\right)[h\left(y\right)-y]$ starting from initial condition $y_0$ for $\omega \left(t\right)\equiv {\omega }_{+}$ and $\omega \left(t\right)\equiv {\omega }_{-}$, respectively. (d) System dynamics subject to random switching control $\omega \left(t\right)={\omega }_{\sigma \left(t\right)}$, which can effectively be described as a random walk with absorbing boundaries.

Figure 4

Clusters formed in executing random switching control for the cell system (8), where the coupling strength is $k_2=8$ and other parameters are $\mathrm{\Delta }T=10$ and ${\omega }_{\pm }=\pm 1$.

Figure 5

(a) Dynamics of three mRNA in the circadian rhythm model with sustained oscillation. (b) Convergence to the biologically significant USS with switching control.

Figure 6

For the Mackey-Glass system with different initial conditions $x\left(t\right)\equiv x_0$ for $t\in (-\tau ,0)$, three controlled and detected USSs.

Figure 7

(a) Dynamics of an uncoupled Duffing oscillator with three equilibria. (b) Chaotic behavior in the weakly coupled Duffing oscillator system for $c_{1,2}=0.1$. Our control method yields all nine unstablse equilibria, marked as red squares. (c) The probabilities of convergence to the nine equilibria. (d) Composition of the subplots in (c), where the surface in (d) corresponds to the highest probability from the subplots in (c).