Phase and frequency linear response theory for hyperbolic chaotic oscillators

Abstract

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase, such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes of frequency are experimentally accessible giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.

Figures (7)

• Figure 1: (a) The mean frequency of the chaotic Roessler oscillator (Eq. \ref{['Eq:RoesslerSystem']}, $a=0.25, c=6.0$) is not a differentiable function of the system parameter $b$ at points of bifurcation. Shown are histograms of return times to the Poincare section $P_{\vartheta_0}$ at $\vartheta_0=\pi/3$ and the mean period (blue line) as functions of $b$. (b) Unstable periodic orbit (solid blue line) of the chaotic Roessler oscillator Eq. \ref{['Eq:RoesslerSystem']} ($a=0.25, b=0.9, c=6.0$) with natural frequency $\omega_0=1.04$. The invariant linear subspaces under system propagation of one period (black polygons) are linear approximations of the UPO's isochrons. The red line is the linear approximation of the UPO's shadow under periodic forcing of $\varepsilon\sin(\Omega t)$ in the $x$-direction. The UPO was found via numerical root finding on a Poincare section, the stable and unstable directions by forward and backward integration, and the shadow was constructed with the method described in Sec.\ref{['sec:shadow']}. With $\varepsilon=0.4$ and $\Omega=1.07$ the shadowing trajectory is synchronized and phase locked to the forcing.
• Figure 2: Three dimensional schematics for the linear dynamics near a point $\vec{x}_0(\varphi)$ of an unperturbed trajectory (light green line). The subspace spanned by the stable and unstable directions (co-variant Lyapunov vectors $\vec{v}^-$ and $\vec{v}^+$) is an isochron (black polygons). The phase sensitivity $\vec{Z}$ is orthogonal to the isochron. A single kick of amplitude and direction $\Delta\vec{x}$ takes the shadow trajectory ($\vec{x}(\varphi)$, dark red line) from a point on the unstable manifold to a point on the stable manifold and advances the phase by $\Delta\varphi = \vec{Z}\cdot\Delta\vec{x}$.
• Figure 3: Numerical integration of chaotic electro-chemical oscillator model Eqs. \ref{['Eq:GaspardOsc']}-\ref{['Eq:GaspardNonlin']} over $400$ time units with $d\varphi=1\times 10^{-3}$. Transients for the convergence of Lyapunov vectors have been discarded. (a) chaotic attractor in the time-delay embedding $(x,y,z) = (E(t),E(t-0.3),E(t-0.6))$ Color coded are small intervals of the optimized phase $\vartheta=\vartheta_\sigma$. Blue shades signify regions of positive PRC and red hues negative values. (b) Chaotic attractor in the original dynamic variables $(E,U,W)$. The color code of the phase intervals is the same as for the corresponding points in (a). In (c) we show the velocity of the geometric proto-phase $\vartheta_0$ (blue lines, left axis) and compare them to the velocity of the optimized phase (orange lines, right axis) with much smaller standard deviation (3.67 vs. 0.03). Both phase velocities are shown as functions of the optimized geometric phase. In subfig. (d) we compare the component of $\vec{Z}({\vec{x}_0})$ in the $E$ direction obtained by the method of Lyapunov vectors (light blue lines) and their average at constant angle $\vartheta$ (large red squares) with frequency response curves obtained from kicking the oscillator in the $E$ direction every time the Poincare section $P_\vartheta$ is crossed after completing one rotation. Up to a kick strength of $\Delta E \le 5\times 10^{-4}$ the curves follow the theoretical prediction via the method of Lyapunov vectors. (e) Local Lyapunov exponents $\lambda^{(+)}$ and $\lambda^{(-)}$ for the Lyapunov vectors in the unstable and stable directions. Both have large deviations in the positive and negative directions, but $\Lambda^{(+)}=\langle\lambda^{(+)}\rangle = 0.07$ (blue dashed line) and $\Lambda^{(+)}=\langle\lambda^{(+)}\rangle = -2.5$ (red dashed line) are rather small. Finally in (f) we show the distance $|\vec{h}|$ of the shadow trajectory which is kicked at optimized geometric phase $\vartheta=0$ with strength $\Delta E = 1\times 10^{-4}$. For larger values of $\Delta E$ the distance of the shadow in linear approximation would increase proportionally.
• Figure 4: Frequency response in the chaotic Roessler system \ref{['Eq:RoesslerSystem']}. (a) Chaotic attractor with color coded small intervals of optimized geometric phase $\vartheta(\vec{x}_0)$. Blue hues indicate negative values of $Z_x$ and red hues positive values. Panels (b-d) show the components of the phase sensitivity $\vec{Z}(\vec{x}_0)$ (thin blue lines) as a function of the optimized geometric phase $\vartheta$, a narrow Gaussian average of these values as red dots, disregarding values of $Z$ larger than three standard deviations, and (white square markers) the linear response of the oscillation period to delta kicks of strength $\varepsilon=0.05$ in the three dynamical variables (b) $x$, (c) $y$ and (d) $q$ at the crossing of a given Poincare section in the optimized geometric phase after each full rotation.
• Figure 5: Hyperbolic activator-inhibitor dynamics (\ref{['Eq:FlipFlop11']}-\ref{['Eq:FlipFlop22']}) of two coupled oscillators with chaotic phase dynamics kuznetsov2007autonomous. (a) log-amplitudes $q_i=\log a_i$ and square amplitudes $a_i^2 = x_i^2+y_i^2$ (shown in inset). (b) $x$ coordinates of the two oscillators as a function of geometric phase $\vartheta$ over 9 periods of the amplitude oscillations. (c) The Poincare map of the angle $\psi_1$ at geometric phase $\vartheta_0=0$ is an expanding circle map. (d) Distance $h=|\vec{h}|$ of the perturbed trajectory from an unperturbed trajectory for log-amplitude $\delta$-Kicks of strength $\varepsilon=0.01$ at geometric phase $\vartheta=0$. The distance after the kick is smaller than before because relaxation in the unstable directions is slower and in that direction the shadowing trajectory is by construction kicked back to the unperturbed trajectory. (e) Component $Z_{q_1}$ and (f) component $Z_{\psi_1}$ of the Lyapunov co-vector $\vec{Z} = \vec{u}^{(0)}$ (thin lines) and period response (\ref{['Eq:FreqShift']}) to kicking the log-amplitude $q_1$ or the angle $\psi_1$ of the first oscillator at a given geometric phase $\vartheta$ (dot markers) with $\varepsilon=0.1$.