A note on finite-time Lyapunov dimension of the Rossler attractor
For the Rössler system we verify Eden’s conjecture on the maximum of local Lyapunov dimension. We compute numerically finite-time local Lyapunov dimensions on the Rössler attractor and embedded unstable periodic orbits. The UPO computation is done by Pyragas time-delay feedback control technique.
I Rössler attractor and Pyragas stabilization of embedded unstable periodic orbits
Consider the following Rössler system Rossler-1976 ()
with arbitrary real parameters . If , then system (1) has the following equilibria:
For some values of parameters system (1) exhibits chaotic behavior. To get a visualization of chaotic attractor one needs to choose an initial point in the basin of attraction of the attractor and observe how the trajectory, starting from this initial point, after a transient process visualizes the attractor: an attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of an equilibrium, otherwise, it is called a hidden attractor LeonovKV-2011-PLA (); LeonovK-2013-IJBC (); LeonovKM-2015-EPJST (); Kuznetsov-2016 (). It was discovered numerically by Rössler that in the phase space of system (1) with parameters , , there exist a chaotic attractor of spiral shape, which is self-excited with respect to both equilibria .
One of the building blocks of chaotic attractor are embedded unstable periodic orbits (UPOs). An effective method for the computation of UPOs is the time-delay feedback control (TDFC) approach, suggested by K. Pyragas Pyragas-1992 () (see also discussions in KuznetsovLS-2015-IFAC (); ChenY-1999 (); CruzVillar-2007 (); LehnertHFGFS-2011 ()). Let be an UPO with period , , satisfying a differential equation
To compute the UPO, we add the TDFC:
Then for the initial point , chosen on the UPO , }, we numerically compute the trajectory of system (4) without the stabilization (i.e. with ) on sufficiently large time interval (see Fig. (b)b). One can see that on the initial small time interval , even without the control, the obtained trajectory traces approximately the ”true” periodic orbit . But for without control the trajectory diverge from and wind on the attractor .
Ii Finite-time Lyapunov dimension and Eden conjecture
For an attractor, an interesting question (Eden-1989-PhD, , p.98) (known as Eden conjecture) is whether the supremum of the local Lyapunov dimensions is achieved on a stationary point or an unstable periodic orbit embedded in the strange attractor. In general, a conjecture on the Lyapunov dimension of self-excited attractor Kuznetsov-2016-PLA (); KuznetsovLMPS-2018 () is that for a typical system the Lyapunov dimension of a self-excited attractor does not exceed the Lyapunov dimension of one of unstable equilibria, the unstable manifold of which intersects with the basin of attraction and visualize the attractor.
Below we follow the concept of the finite-time Lyapunov dimension Kuznetsov-2016-PLA (); KuznetsovLMPS-2018 (), which is convenient for carrying out numerical experiments with finite time. The finite-time local Lyapunov dimension Kuznetsov-2016-PLA (); KuznetsovLMPS-2018 () can be defined via an analog of the Kaplan-Yorke formula with respect to the set of finite-time Lyapunov exponents:
where . Then the finite-time Lyapunov dimension (of dynamical system generated by (3) on compact invariant set ) is defined as
The Douady–Oesterlé theorem DouadyO-1980 () implies that for any fixed the finite-time Lyapunov dimension, defined by (6), is an upper estimate of the Hausdorff dimension: . The best estimation is called the Lyapunov dimension Kuznetsov-2016-PLA ()
For the Rössler attractor the Lyapunov dimension was estimated as FroehlingCFPS-1981 (), SanoS-1985 (), Sprott-2003 (); Fuchs-2013 (), and AwrejcewiczKEDBK-2018 ()); see also KuznetsovMV-2014-CNSNS (); SprottL-2017 ().
Below we use the adaptive algorithm KuznetsovLMPS-2018 () for the computation of the finite-time Lyapunov dimension and exponents. We compute: maximum of the finite-time local Lyapunov dimensions at the points of grid filling the attractor , i.e. ; finite-time Lyapunov dimensions for the stabilized UPO with periods .
The comparison of the obtained values of and computed along the stabilized UPO and the trajectory without stabilization gives us the following results. On the initial part of the time interval, one can indicate the coincidence of these values with a sufficiently high accuracy. For the period-1 UPO and for the unstabilized trajectory the largest Lyapunov exponents coincide up to the 5th decimal place inclusive on the interval . After the difference in values becomes significant and the corresponding graphics diverge in such a way that the part of the graph corresponding to the unstabilized trajectory is lower than the part of the graph corresponding to the UPO (see Fig. (b)b).
The equilibria has simple eigenvalues and, thus, we have .
The period-1 UPO with period has the following multipliers: , , . Thus, for the local Lyapunov dimension of the UPO we obtain .
In this note we have confirmed the Eden conjecture for the Rössler system (1) and obtained the following relations between the Lyapunov dimensions:
Concerning the time of integration, remark that while the time series obtained from a physical experiment are assumed to be reliable on the whole considered time interval, the time series produced by the integration of mathematical dynamical model can be reliable on a limited time interval only due to computational errors (caused by finite precision arithmetic and numerical integration of ODE). Thus, in general, the closeness of the real trajectory and the corresponding pseudo-trajectory calculated numerically can be guaranteed on a limited short time interval only. However, for two different long-time pseudo-trajectories and visualizing the same attractor, the corresponding finite-time LEs can be, within the considered error, similar due to averaging over time and similar sets of points and . At the same time, the corresponding real trajectories may have different LEs, e.g. may correspond to an unstable periodic trajectory which is embedded in the attractor and does not allow one to visualize it.
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