Time-delay identification from chaotic time series via statistical complexity measures based on ordinal pattern transition networks

Abstract

In this paper, a novel time-delay parameter identification method is proposed based on nonlinear time series analysis combined with network theory. This method accurately reveals the intrinsic time-delay characteristics of the underlying system dynamics. Time-delay parameters are identified from chaotic time series by using two statistical complexity measures defined by two normalized matrices encoding ordinal pattern transition networks of the time series. The proposed method is straightforward to apply and has two prime advantages: it is robust to time series contaminated by dynamical or observational noise, and it is well suited for handling relatively short time series. It increases the upper bound of the possible range of applicable (dynamical) noise intensities by at least two orders of magnitude versus the permutation–information–theory approach. It can also detect two time-delay parameters for relatively short time series in a two-time-delay system, which the delayed mutual information cannot do.

Appendices

Appendix A: The iteration formula for simulation

The iteration formula of the simulation time series of Eq. (1) by using the modified Euler’s method can be expressed as follows:

$$ \overline{x}_{i + 1} = x_{i} + h[ - x_{i} + \beta {\text{sin}}^{2} \left( {x_{i - C} { }{-}{ }\varphi_{0} } \right)] $$

(A.1)

$$ x_{i + 1} = x_{i} + \frac{h}{2}\left[ {\left( { - x_{i} + \beta {\text{sin}}^{2} \left( {x_{i - C} { } - { }\varphi_{0} } \right)} \right) + \left( { - \overline{x}_{i + 1} + \beta {\text{sin}}^{2} \left( {x_{i + 1 - C} { } - { }\varphi_{0} } \right)} \right)} \right] $$

(A.2)

where C = τ/h is a nonnegative integer.

The iteration formula of the simulation time series of Eq. (2) by using the modified Euler’s method can be expressed as follows:

$$ \overline{x}_{i + 1} = x_{i} + h\left[ { - x_{i} + \beta {\text{sin}}^{2} \left( {x_{i - C} { } - { }\varphi_{0} } \right)} \right] + h\theta_{i} $$

(A.3)

$$ x_{i + 1} = x_{i} + \frac{h}{2}\left[ {\left( { - x_{i} + \beta {\text{sin}}^{2} \left( {x_{i - C} { } - { }\varphi_{0} } \right)} \right) + \left( { - \overline{x}_{i + 1} + \beta {\text{sin}}^{2} \left( {x_{i + 1 - C} { } - { }\varphi_{0} } \right)} \right)} \right] + \theta_{i} $$

(A.4)

where C = τ/h is a nonnegative integer, and the {${\theta }_{i}$} is a Gaussian white noise series with the noise intensity D.

The generation step of the {${\theta }_{i}$} can be described as: first and foremost, the independent identically distributed standard normal random number {${\xi }_{i}$} is obtained via the random number generator. Then, the {${\theta }_{i}$} can be generated by transforming the {${\xi }_{i}$} into formula ${\theta }_{i}$ = $\sqrt{D/h}{\xi }_{i}$.

The iteration formula of the simulation time series of Eq. (3) by using the modified Euler’s method can be expressed as follows:

$$ \overline{x}_{i + 1} = x_{i} + h\left[ { - x_{i} + \beta {\text{sin}}^{2} \left( {x_{{i - C_{1} }} + { }\varphi_{0} \left( {1 + { }k_{p} x_{{i - C_{2} }} } \right)} \right)} \right] $$

(A.5)

$$ x_{i + 1} = x_{i} + \frac{h}{2}\left[ {\left( { - x_{i} + \beta {\text{sin}}^{2} \left( {x_{{i - C_{1} }} + { }\varphi_{0} \left( {1 + { }k_{p} x_{{i - C_{2} }} } \right)} \right)} \right) + \left( { - \overline{x}_{i + 1} + \beta {\text{sin}}^{2} \left( {x_{{i + 1 - C_{1} }} + { }\varphi_{0} \left( {1 + { }k_{p} x_{{i + 1 - C_{2} }} } \right)} \right)} \right)} \right] $$

(A.6)

Where β = 3.5, ${\varphi }_{0}$ = 3.5, ${k}_{p}$ = 0.43. ${C}_{1}$ = ${\tau }_{1}$/h and ${C}_{2}$ = ${\tau }_{2}$/h are nonnegative integers, where ${\tau }_{1}$ = 100 and ${\tau }_{2}$ = 77.

Appendix B: Identification results for two other typical time-delay systems

To demonstrate the universality of the proposed methodology, identification results for two representative time-delay systems are provided. The first system is the following Mackey–Glass equation [1]:

$$ \frac{{{\text{d}}x\left( t \right)}}{{{\text{d}}t}} = - x\left( t \right) + \frac{{\beta x\left( {t {-}{ }\tau } \right)}}{{1 + x^{c} \left( {t {-} \tau } \right)}} $$

(B.1)

System parameters are set to β = 2, c = 10. The time-delay parameter is set to τ = 60. The initial function is x(t) = 1, t ∊ [-τ, 0]. The second system is the following time-delay Duffing equation [72]:

$$ \frac{{{\text{d}}x\left( t \right)}}{{{\text{d}}t}} = y\left( t \right) $$

(B.2.1)

$$ \frac{{{\text{d}}y\left( t \right)}}{{{\text{d}}t}} = - \xi y\left( t \right) - \omega x\left( t \right) - \mu x^{3} \left( t \right) - \beta x\left( {t - { }\tau } \right) + f{\text{cos}}(\Omega t) $$

(B.2.2)

System parameters are set to ξ = 0.2, ω = -1, μ = 0.8, β = -0.8, f = 0.5, Ω = 1. The time-delay parameter is set to τ = 100. Initial functions are x(t) = 1 and y(t) = 1, t ∊ [-τ, 0].

For these two systems, the simulation method used here is the same as that used in Sect. 3. Note that the y-coordinate time series is employed for the time-delay Duffing system to construct the corresponding OPTN. Identification results of the time-delay parameter τ by utilizing the proposed quantifiers with the embedding dimension M = 4 are shown in Fig. 

Fig. 10

(Color online) a to d show the curve of the proposed quantifiers as a function of the embedding delay ${\tau }_{e}$ for the Mackey–Glass system. e to h show the curve of the proposed quantifiers as a function of the embedding delay ${\tau }_{e}$ for the time-delay Duffing system

10a–h. It can be clearly observed that each curve has a significant extremum at ${\tau }_{e}$ = τ/δh (for the Mackey–Glass system ${\tau }_{e}$ = 120 and for the time-delay Duffing system ${\tau }_{e}$ = 200).