# The fastest, simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems

- 298 Downloads

## Abstract

Classical method of Lyapunov exponents spectrum estimation for a *n*-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms. In this paper, we have shown that using a new, simplified method, it is possible to estimate full spectrum of *n* Lyapunov exponents by integration of \((n-1)\) perturbations only. In particular, it is enough to integrate just one perturbation to obtain two largest Lyapunov exponents, which enables to search for hyperchaos. Moreover, in the presented algorithm, only very basic mathematical operations such as summation, multiplication or division are applied, which boost the efficiency of computations. All these features together make the new method faster than any other known by the authors if the order of the system under consideration is low. Correctness the method has been tested for three examples: Lorenz system, Duffing oscillator and three Duffing oscillators coupled in the ring scheme. Moreover, efficiency of the method has been confirmed by two practical tests. It has been revealed that for low-order systems, the presented method is faster than any other known by authors.

## Keywords

Lyapunov exponents Spectrum Continuous systems Estimation method Nonlinear dynamics## 1 Introduction

Depending on a dynamical system type and a kind of information that is useful for its investigations, different types of invariants characterizing system dynamics are applied. For instance, one may use Kolmogorov entropy [1, 2], correlation dimension [3, 4], Lyapunov energy function [5] to determine the stability of solutions and complexity of system dynamics [6]. However, in order to predict the influence of perturbations on solution of a system, Lyapunov exponents (LEs) are one of the most commonly applied tools. That is because these exponents determine the exponential convergence or divergence of trajectories that start close to each other. The existence of such numbers has been proved by Oseledec theorem [7], but the first numerical study of the system’s behavior using Lyapunov exponents had been done by Henon and Heiles [8], before the Oseledec theorem publication. The most important algorithms for calculating Lyapunov exponents for continuous systems have been developed by Benettin et al. [9] and Shimada and Nagashima [10], later improved by Benettin et al. [11] and Wolf [12]. In order to estimate Lyapunov exponents from a scalar time series, the Takens procedure [13] can be applied. This approach can be utilized also in the cases when discontinuities or time delays exist in the analyzed system. Numerical algorithms for such estimation have been developed by Wolf et al. [14], Sano and Sawada [15] and later improved by Eckmann et al. [16], Rosenstein et al. [17] and Parlitz [18]. Alternative method applicable to systems with discontinuities or time delays, based on synchronization phenomena, was elaborated by Stefanski [19, 20, 21, 22].

Nowadays, Lyapunov exponents are employed in many different areas of scientific research such as: materials [23, 24], electric power systems [25], non-continuous systems [20, 22, 26, 27, 28, 29], systems with time delay [21], aerodynamics [30], time series analysis [31, 32, 33, 34], optimal control [35, 36], chaotic encryption and secure communication [37, 38], multi-objective optimization [39], parametric oscillations and fluctuating parameters [19, 40], neuronal models investigations [41].

Thus, there is still need to elaborate fast and simple methods of LE calculation. Recently, we have presented our simple and effective method of estimation of the largest Lyapunov exponent (LLE) from the perturbation vector and its derivative dot product. The method is based on a clear, geometrical reasoning. Its application involve only basic mathematical operations such as summing, multiplying, dividing. The method can be applied in different aspects of the nonlinear systems control. We investigated continuous systems [42], synchronization phenomena detection [43], time series in control systems [44, 45, 46].

*n*Lyapunov exponents by integration of \(\left( {n-1} \right) \) perturbations only. The algorithm has been tested for three examples: Lorenz system, Duffing oscillator and three Duffing oscillators coupled in the ring scheme. Selection of the exemplary systems can be justified by the fact that they exhibit different types of dynamical behavior depending on the analyzed parameters. Therefore, correctness of the method has been confirmed for periodic, quasiperiodic and chaotic trajectories [47]. Computation times and convergence rates have been compared with the classical method [47] that involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms. The method presented in [47] uses the same approach as algorithms described in [9, 11]. To authors’ best knowledge, no algorithms of LE spectrum estimation for continuous-time, smooth dynamical systems faster than [9, 10, 11, 47] have been published. In this paper, it has been revealed that application of our new algorithm increases the efficiency of the calculations compared to the classical method [47] when the order of the system is low: 2, 4 or 6. Such acceleration has been achieved not only due to integration of \(\left( {n-1} \right) \) perturbations instead of

*n*, but also owing to the simplicity of the algorithm, which involves only the basic mathematical operations: addition, multiplication, division. Compared to the classical method, calculation of logarithms and normalization of perturbations after each iteration can be omitted. Therefore, authors claim that the method presented in this paper is the fastest one in the assumed range of applications if the order of the system is low enough.

## 2 The method

*t*is the time and \({{\varvec{f}}}\) is a vector field that (in general) depends on \({{\varvec{x}}}\) and

*t*. Evolution of any perturbation \({{\varvec{z}}}\) in such system can be found from the equation:

According to [47], each LE \(\lambda _i \) is the average rate of perturbation contraction or expansion in a particular subspace near a particular limit set. The LLE—\(\lambda _1 \)—is the average rate of perturbation length change in (almost) whole space in which system (1) is defined. Let this whole space be named \(W_1 \).

*n*, there are

*n*nested subspaces:

*i*-th LE can be obtained from evolution of a perturbation in the subspace \(W_i \).

The classical method of LE spectrum estimation involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms [47]. In this approach, *n* perturbations \({{\varvec{z}}}_\mathbf{1} , {{\varvec{z}}}_\mathbf{2} , \ldots , {{\varvec{z}}}_{{\varvec{n}}}\) are integrated. Orthonormalization procedure assures that each perturbation \({{\varvec{z}}}_{{\varvec{i}}}\) evolves in the subspace \(W_i \). Values of LE are obtained by calculating natural logarithm of perturbations lengths after subsequent integration steps and averaging these values over time. The crucial fact is that in order to calculate whole spectrum of *n* LE, integration of *n* perturbations is necessary.

In the presented method, complexity of calculations is reduced. In order to estimate the whole spectrum of *n* LE, it is enough to integrate \(\left( {n-1} \right) \) perturbations. For clarity, in the next paragraph, the case of a second-order system is discussed. Authors demonstrate how integration of only one perturbation \({{\varvec{z}}}_\mathbf{1}\) enables to estimate both LE: \(\lambda _1 , \lambda _2 \). The subsequent paragraph covers the general case, i.e., estimation of a full LE spectrum of a \(n\mathrm{th}\)-order system using only \(\left( {n-1} \right) \) integrated perturbations.

Let us assume that for a \(2\mathrm{nd}\)-order system, only one perturbation \({{\varvec{z}}}_\mathbf{1}\) is integrated. Obviously, using formula (3), LLE can be calculated. As \({{\varvec{z}}}_\mathbf{1}\) evolves according to (2), it tends to line up with the direction in which evolution of the perturbation is governed mainly by the value of \(\lambda _1 \), whereas the terms connected with \(\lambda _2 \) become negligible [47]. Therefore, throughout its evolution, \({{\varvec{z}}}_\mathbf{1}\) tends to a direction orthogonal to \(W_2 \). Let us also take any random vector \({{\varvec{z}}}_\mathbf{2}\), the length of which is nonzero and which is not parallel to \({{\varvec{z}}}_\mathbf{1}\) (Fig. 1).

*n*. Suppose that \(\left( {n-1} \right) \) perturbations \({{\varvec{z}}}_\mathbf{1} , {{\varvec{z}}}_\mathbf{2} , \ldots , {{\varvec{z}}}_{{{\varvec{n}}}-\mathbf{1}} \) are integrated

^{1}. Their time derivatives can be found from formula (2). As \({{\varvec{z}}}_\mathbf{1}\) evolves, it tends to line up with the direction in which evolution of the perturbation is governed mainly by the value of \(\lambda _1 \), whereas the terms connected with subsequent LE \(\lambda _2 , \lambda _3 , \ldots , \lambda _n \) become negligible [47]. Therefore, throughout its evolution, \({{\varvec{z}}}_\mathbf{1}\) tends to a direction orthogonal to \(W_2 \). The component \({\tilde{{{\varvec{z}}}}}_\mathbf{2}\) of the perturbation \({{\varvec{z}}}_\mathbf{2}\), which is perpendicular to \({{\varvec{z}}}_\mathbf{1}\), can be obtained by means of formula (5). To a good approximation \({\tilde{{{\varvec{z}}}}}_\mathbf{2} \in W_2 \). Moreover, \({\tilde{{{\varvec{z}}}}}_\mathbf{2}\) tends to a direction which is orthogonal to \(W_3 \). Consequently, vectors \(\tilde{\mathbf{z}}_{\mathbf{3}} , \tilde{\mathbf{z}}_\mathbf{4} , \ldots , \tilde{\mathbf{z}}_{{{\varvec{n}}}-\mathbf{1}}\) can be calculated using the following orthogonalization expression:

## 3 Numerical simulations

Simulation programs were written in C++ using CodeBlocks 16.01. Graphs and diagrams were built with *matplotlib* library as a tool of Python programming language. In order to integrate the system of differential Eqs. (1)–(2), Runge-Kutta method of the fourth order was used. The integration step was chosen experimentally for each system.

In order to estimate the whole spectrum of Lyapunov exponents, the method requires to simulate \(\left( {n-1} \right) \) perturbations, where *n* is the order of the system. For example, for the second-order non-autonomous Duffing oscillator with external excitation [42], one perturbation was calculated. For the third-order autonomous system of Lorenz equations [48], two disturbances were observed. For three Duffing systems coupled in a ring scheme [50], which together constitute a system of the \(6\mathrm{th}\) order, five perturbations were integrated. After each integration step, subsequent perturbations were normalized and orthogonalized according to Eqs. (5), (7).

The simulation program remembered subsequent values of \(\lambda _i^*\) in *n* buffers of a fixed capacity. When buffers were full, the program calculated standard deviation of all the values in each of *n* buffers. If the standard deviation of values in any buffer was too high, buffers were cleared and computations were continued. Otherwise, if the value of standard deviation for each buffer was below a fixed threshold, the calculations were terminated. The final value of \(\lambda _i \) returned by the program was equal to the average of all values \(\lambda _i^*\) remembered in the *i*-th buffer.

## 4 Results of numerical simulations

*r*was used as the control parameter in the range from 0 to 100. Initial conditions for \(r=0\) were set as a random three-element vector of magnitude 1. Initial perturbations were also chosen randomly and normalized to 1.

*z*in system (9) is shown in Fig. 2. The corresponding graph of LE spectrum calculated using the new method is depicted in Fig. 3, whereas the LE spectrum graph obtained by means of the classical method is shown in Fig. 4.

The bifurcation diagram presented in Fig. 2, together with the LE diagram presented in Figs. 3, 4, demonstrates that the Lorenz system (9) is periodic for all the values of *r* smaller or equal to 24.7, whereas it is chaotic for most values of *r* larger or equal to 24.8. However, for some values of *r* above 24.8, periodic windows exist. Detailed information on different kinds of limit sets in dynamical systems can be found in [47]. It can be noticed that the graphs in Fig. 3 (the new method) and in Fig. 4 (the classical method) are in complete agreement. Moreover, both graphs are consistent with [48], which confirms the correctness of the algorithm presented in this paper.

*q*was used as the control parameter in the range from 0.5 to 1.8 [42]. In the simulation, a random vector of length 1 was used as initial condition. Initial perturbations were also random and normalized to 1.

*x*of the Duffing Eq. (11) is shown in Fig. 5. The corresponding graph of LE spectrum calculated using the new method is depicted in Fig. 6. The LE spectrum graph obtained by the classical method is presented in Fig. 7. In both cases, red line corresponds to the largest LE, while blue line corresponds to the second Lyapunov exponent of system (11).

Figures 5, 6 and 7 show that the Duffing system is chaotic for most values of the parameter *q* in ranges from 0.693 to 0.738 and from 1.237 to 1.673. However, some periodic windows exist in these ranges too. For the values of *r* between 0.5 and 0.692, from 0.739 to 1.236 and from 1.674 to 1.8, the system is periodic. For the Duffing system, the generalized divergence of the flow is constant and equal to \(-\beta \) [49]. Sum of Lyapunov exponents has to be equal to the generalized divergence of the flow. Therefore, graphs of the first and second Lyapunov exponents should be symmetrical with respect to horizontal line \(\lambda =-\frac{\beta }{2}\). This fact was confirmed in Fig. 6 and in Fig. 7.

*k*was used as the control parameter in the range from 0 to 4. Initial conditions for \(k=0\) were set as a random six-element vector of magnitude 1. Initial perturbations were also chosen randomly and normalized to 1. Please note that names of parameters are different in formulas (11) and (13). This is due to the fact that results of both simulations can be verified in two different sources. Results of calculations for system (11) can be checked in [42], whereas system (13) can be verified in [50].

Figures 8, 9 and 10 demonstrate how dynamics of three Duffing systems (13) changes as the coupling strength increases [50]. For the values of *k* smaller than 0.24, one can notice that all the LEs are negative and the system tends to a stable fixed point. For the values of *k* between 0.24 and 0.38, one LE is equal to 0, which corresponds to a periodic solution. In the range of parameter *k* from 0.38 to 1.79, more than one LEs are zero and a quasiperiodic solution is obtained. Transition to chaos takes place for *k* equal to 1.79, followed by transition to hyperchaos as coupling coefficient reaches 1.9. Please note that the full spectrum of six LEs was obtained by integration of five perturbation vectors. However, if the task is to just detect chaos and hyperchaos, only one perturbation is necessary when our new method is applied, whereas integration of two perturbations is required if the classical method is used.

## 5 Computation time comparisons

In the previous sections, the novel method has been explained and results of its application have been demonstrated. It has been confirmed that the novel method requires integration of only \(\left( {n-1} \right) \) perturbations to estimate *n* Lyapunov exponents, which is expected to increase efficiency of calculations. Moreover, formulae (3) and (8), which are used for LE estimation in the new method, require only the most basic mathematical calculations—addition, multiplication and division. In such a manner, calculation of perturbation’s norm logarithm, which is necessary in the classical method, can be avoided. Therefore, there are two possible reasons for efficiency of the novel method. Firstly, size of the problem is reduced, because only \(\left( {n-1} \right) \) perturbations are integrated. However, computation speed gain due to this feature may vary with the order of the system. Secondly, application of formulae (3) and (8) instead of estimation of the logarithm of the perturbation norm can additionally shorten the computation time.

Efficiency of both approaches has been tested by a C++ program. The program selects \(2*10^{7}\) vectors at random. Then, it checks how much time it takes to estimate natural logarithm of the norm of each of these vectors, just as it is done in the classical method of LE estimation [47]. Afterward, formula (3) is applied for each vector from the same set and time of these operations is also measured. In such a manner, efficiency of both methods of LE estimation can be compared. This test has been executed 100 times for different vector dimensions. The average, the minimum and the maximum computation time has been recorded for both methods.

However, an important issue arises: Efficiency of the novel method of LE estimation can be largely influenced by estimation of the perturbation’s derivative, which is required by the new method. If the Jacobi matrix \({{\varvec{U}}}\left( {{\varvec{x}}} \right) \) contains only few nonzero elements, then calculation of the perturbation’s derivative according to formula (2) is fast. When most elements of \({{\varvec{U}}}\left( {{\varvec{x}}} \right) \) are nonzero, then calculation of the perturbation’s derivative is slower. Therefore, two separate cases have been analyzed. In the best scenario, only one element in each row of \({{\varvec{U}}}\left( {{\varvec{x}}} \right) \) is nonzero. In the worst case, all elements of \({{\varvec{U}}}\left( {{\varvec{x}}} \right) \) are nonzero. Results for both cases with different vector dimensions are presented in Figs. 11 and 12. These figures represent the best case and the worst case, respectively. In each of these graphs, the computation time ratio is obtained by dividing computation time of formula (3) by the average time of calculations with the classical method, i.e., when the natural logarithm of the vector’s norm is estimated. Therefore, value 1 in the graph means that the computation time for both methods is equal. Values below 1 show that the new method is faster, and values above 1 indicate that the novel method is slower.

Now, as the influence of using the simplified formulae (3) and (8) has been investigated, overall efficiency of the novel method can be checked on a practical example. To do so, the spectrum of Lyapunov exponents of Duffing systems coupled in ring scheme (13) has been estimated with both methods for different number of oscillators in the ring, from 1 to 5. Therefore, systems of orders \(2, 4, \ldots , 10\) have been investigated. The same coupling strength \(k=3.0\) has been used in all tests. For each number of oscillators, the Lyapunov exponents spectrum has been estimated 100 times using both methods. The orthonormalization process [47] has been performed after each iteration of both methods. The average, minimum and maximum computations times have been recorded. The results are presented in Fig. 13. The computation time ratio is obtained by dividing the calculations time of the novel method by the average time of calculations of the classical method. Therefore, just as in previous graphs, values below 1 indicate that the novel method is faster, whereas values above 1 show that the classical method performs better.

Figure 13 confirms that the novel method, presented in this paper, is faster than the classical method [47] in each test case, as long as order of system (13) is smaller or equal to 4. For the system with three oscillators and order 6, the novel method is faster in the average, but it is not faster in each trial. When more than three oscillators are coupled together, the classical method performs better.

## 6 Conclusions

The presented article shows that from dot products of perturbations vectors and their derivatives, one can easily extract the full spectrum of Lyapunov exponents. We have shown that full spectrum of LE for a system of order *n* can be obtained by integrating only \(\left( {n-1} \right) \) perturbations. Moreover, due to the fact that our method is based on very unsophisticated computations involving only basic mathematical operations such as summing, subtracting, multiplying, dividing, it is more efficient than the classical one when the order of the system is low enough. We have presented the theoretical description showing the mathematical simplicity of the method. The correctness and effectiveness of our method has been confirmed by simulations of autonomous and non-autonomous nonlinear dynamical systems of different orders. These results have been compared with the ones obtained using the classical algorithm. The efficiency of the new method has been compared to the method that involves the calculation of perturbations lengths logarithms and Gram–Schmidt orthonormalization. Firstly, it has been shown that application of the new, simple formulas for LE estimation is faster than calculation of the natural logarithm of perturbation’s norm for systems of dimension smaller or equal to 4, even if the Jacobi matrix of the system is dense. It has been revealed that if the Jacobi matrix is sparse, then the novel formulas can perform better even for systems of the dimension 10 or higher. Secondly, efficiency of the novel method has been verified for Duffing systems coupled in the ring scheme. It turned out that the novel method is in average faster if the dimension of the system is smaller or equal to 6. Obviously, efficiency of both methods may vary depending on properties of the system under consideration. Moreover, implementation of the method may influence the effectiveness. However, taking into account the obtained results, it can be stated that for low-dimensional, continuous-time, smooth systems, the presented method of LE spectrum estimation is the fastest one. The next step of the method’s development can be considered in estimation of LE in systems with discontinuities, with time delays and others.

## Footnotes

- 1.
In order to maintain orthogonality of the integrated perturbations, before each integration step the following substitution is performed: \({{\varvec{z}}}_{{\varvec{k}}} :={\tilde{{{\varvec{z}}}}}_{{\varvec{k}}} , k=2, 3, \ldots , \left( {n-1} \right) \). The orthogonalized perturbations \({\tilde{{{\varvec{z}}}}}_{{\varvec{k}}}\) are obtained from formula (7).

## Notes

### Acknowledgements

This study has been supported by Polish Ministry of Science and Higher Education under the program “Diamond Grant,” Project No. D/2013 019743. This study has been supported by Polish National Centre of Science (NCN) under the program MAESTRO: “Multi-scale modelling of hysteretic and synchronous effects in dry friction process,” Project No. 2012/06/A/ST8/00356.

## References

- 1.Bennettin, G., Froeschle, C., Scheidecker, J.P.: Kolmogorov entropy of a dynamical system with increasing number of degrees of freedom. Phys. Rev. A
**19**, 2454–2460 (1979)MathSciNetCrossRefGoogle Scholar - 2.Benettin, G., Galgani, L., Strelcyn, J.M.: Kolmogorov entropy and numerical experiment. Phys. Rev. A
**14**, 2338–2345 (1976)CrossRefGoogle Scholar - 3.Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett.
**50**, 346–349 (1983)MathSciNetCrossRefGoogle Scholar - 4.Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos an Introduction to Dynamical Systems. Springer, New York (2000)zbMATHGoogle Scholar
- 5.Zhang, Y., Chen, D., Guo, D., Liao, B., Wang, Y.: On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn.
**79**, 983–1003 (2015)MathSciNetCrossRefGoogle Scholar - 6.Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.
**57**, 617–656 (1985)MathSciNetCrossRefGoogle Scholar - 7.Oseledec, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc.
**19**, 197–231 (1968)MathSciNetGoogle Scholar - 8.Henon, M., Heiles, C.: The applicability of the third integral of the motion: some numerical results. Astron. J.
**69**, 77 (1964)CrossRefGoogle Scholar - 9.Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, part I: theory. Meccanica
**15**, 9–20 (1980)CrossRefGoogle Scholar - 10.Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys.
**61**(6), 1605–1616 (1979)MathSciNetCrossRefGoogle Scholar - 11.Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, part II: numerical application. Meccanica
**15**, 21–30 (1980)CrossRefGoogle Scholar - 12.Wolf, A.: Quantifying chaos with Lyapunov exponents. In: Holden, V. (ed.) Chaos, pp. 273–290. Manchester University Press, Manchester (1986)Google Scholar
- 13.Takens, F.: Detecting strange attractors in turbulence. Lect. Notes Math.
**898**, 366 (1981)MathSciNetCrossRefGoogle Scholar - 14.Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D
**16**, 285–317 (1985)MathSciNetCrossRefGoogle Scholar - 15.Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Lett.
**55**, 1082–1085 (1985)MathSciNetCrossRefGoogle Scholar - 16.Eckmann, J.P., Kamphorst, S.O., Ruelle, D., Ciliberto, S.: Lyapunov exponents from a time series. Phys. Rev. Lett.
**34**(9), 4971–4979 (1986)Google Scholar - 17.Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D
**65**(1–2), 117–134 (1993)MathSciNetCrossRefGoogle Scholar - 18.Parlitz, U.: Identification of true and spurious Lyapunov exponents from time series. Int. J. Bifurc. Chaos
**2**(1), 155–165 (1992)MathSciNetCrossRefGoogle Scholar - 19.Stefanski, A.: Lyapunov exponents of the systems with noise and fluctuating parameters. J. Theor. Appl. Mech.
**46**(3), 665–678 (2008)Google Scholar - 20.Stefanski, A.: Estimation of the largest Lyapunov exponent in systems with impacts. Chaos Solitons Fractals
**11**(15), 2443–2451 (2000)MathSciNetCrossRefGoogle Scholar - 21.Stefanski, A., Dabrowski, A., Kapitaniak, T.: Evaluation of the largest Lyapunov exponent in dynamical systems with time delay. Chaos Solitons Fractals
**23**, 1651–1659 (2005)MathSciNetCrossRefGoogle Scholar - 22.Stefanski, A., Kapitaniak, T.: Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos Solitons Fractals
**15**, 233–244 (2003)MathSciNetCrossRefGoogle Scholar - 23.Iwaniec, J., Uhl, T., Staszewski, W.: Detection of changes in cracked aluminium plate determinism by recurrence analysis. Nonlinear Dyn.
**70**(1), 125–140 (2012)CrossRefGoogle Scholar - 24.Rybaczuk, M., Aniszewska, D.: Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems. Nonlinear Dyn.
**54**, 345–354 (2008)MathSciNetCrossRefGoogle Scholar - 25.Wadduwage, D.P., Qiong, Wu C., Annakkage, U.D.: Power system transient stability analysis via the concept of Lyapunov Exponents. Electric Power Syst. Res.
**104**, 183–192 (2013)CrossRefGoogle Scholar - 26.Serweta, W., Okolewski, A., Blazejczyk-Okolewska, B., Czolczynski, K., Kapitaniak, T.: Mirror hysteresis and Lyapunov exponents of impact oscillator with symmetrical soft stops. Int. J. Mech. Sci.
**101–102**, 89–98 (2015)CrossRefGoogle Scholar - 27.Serweta, W., Okolewski, A., Blazejczyk-Okolewska, B.: Lyapunov exponents of impact oscillators with Hertz’s and Newton’s contact models. Int. J. Mech. Sci.
**89**, 194–206 (2014)CrossRefGoogle Scholar - 28.Lamarque, C.H., Malasoma, J.M.: Analysis of nonlinear oscillations by wavelet transform: Lyapunov exponents. Nonlinear Dyn.
**9**, 333–347 (1996)MathSciNetCrossRefGoogle Scholar - 29.Yue, Y., Xie, J., Gao, X.: Determining Lyapunov spectrum and Lyapunov dimension based on the Poincare map in a vibro-impact system. Nonlinear Dyn.
**69**, 743–753 (2012)MathSciNetCrossRefGoogle Scholar - 30.Hu, D.L., Huang, Y., Liu, X.B.: Moment Lyapunov exponent and stochastic stability of binary airfoil driven by non-Gaussian colored noise. Nonlinear Dyn.
**70**, 1847–1859 (2012)MathSciNetCrossRefGoogle Scholar - 31.Yang, C., Wu, C.Q., Zhang, P.: Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping. Nonlinear Dyn.
**69**, 1493–1507 (2012)MathSciNetCrossRefGoogle Scholar - 32.Yang, C., Wu, C.Q.: On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping–a case study. Nonlinear Dyn.
**59**, 239–257 (2010)MathSciNetCrossRefGoogle Scholar - 33.Sun, Y., Wu, C.Q.: A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability. Nonlinear Dyn.
**70**, 1689–1708 (2012)MathSciNetCrossRefGoogle Scholar - 34.Yang, C., Wu, C.Q.: A robust method on estimation of Lyapunov exponents from a noisy time series. Nonlinear Dyn.
**64**, 279–292 (2011)MathSciNetCrossRefGoogle Scholar - 35.Zhu, W.Q.: Feedback stabilization of quasi nonintegrable Hamiltonian systems by using Lyapunov exponent. Nonlinear Dyn.
**36**(2), 455–470 (2004)MathSciNetCrossRefGoogle Scholar - 36.Zhu, W.Q., Huang, Z.L.: Stochastic stabilization of quasi-partially integrable Hamiltonian systems by using Lyapunov exponent. Nonlinear Dyn.
**33**(2), 209–224 (2003)MathSciNetCrossRefGoogle Scholar - 37.Li, C., Wang, J., Hu, W.: Absolute term introduced to rebuild the chaotic attractor with constant Lyapunov exponent spectrum. Nonlinear Dyn.
**68**(4), 575–587 (2012)MathSciNetCrossRefGoogle Scholar - 38.Li, S.Y., Huang, S.C., Yang, C.H.: Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling. Nonlinear Dyn.
**69**(3), 805–816 (2012)MathSciNetCrossRefGoogle Scholar - 39.Fraga, L.G., Tlelo-Cuautle, E.: Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dyn.
**76**, 1503–1515 (2014)CrossRefGoogle Scholar - 40.Rong, H., Meng, G., Wang, X.: Invariant measures and Lyapunov exponents for stochastic Mathieu system. Nonlinear Dyn.
**30**(4), 313–321 (2002)MathSciNetCrossRefGoogle Scholar - 41.Soriano, D.C., Fazanaro, F.I., Suyama, R.: A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model. Nonlinear Dyn.
**67**(1), 413–424 (2012)CrossRefGoogle Scholar - 42.Dabrowski, A.: Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product. Nonlinear Dyn.
**67**(1), 283–291 (2012)MathSciNetCrossRefGoogle Scholar - 43.Dabrowski, A.: The largest transversal Lyapunov exponent and master stability function from the perturbation vector and its derivative dot product (TLEVDP). Nonlinear Dyn.
**69**(3), 1225–1235 (2012)MathSciNetCrossRefGoogle Scholar - 44.Balcerzak, M., Dabrowski, A., Kapitaniak, T., Jach, A.: Optimization of the control system parameters with use of the new simple method of the largest Lyapunov exponent estimation. Mech. Mech. Eng.
**17**(3), 225–239 (2013)Google Scholar - 45.Pijanowski, K., Dabrowski, A., Balcerzak, M.: New method of multidimensional control simplification and control system optimization. Mech. Mech. Eng.
**19**(2), 127–139 (2015)Google Scholar - 46.Dabrowski, A.: Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation. Nonlinear Dyn.
**78**(3), 1601–1608 (2014)MathSciNetCrossRefGoogle Scholar - 47.Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989)CrossRefGoogle Scholar
- 48.Moghtadaei, M., Hashemi Golpayenagi, M.R.: Complex dynamic behaviors of the complex Lorenz system. Sci. Iran.
**19**(3), 733–738 (2012)CrossRefGoogle Scholar - 49.Shin, K., Hammond, J.K.: The instantaneous Lyapunov exponent and its application to chaotic dynamical systems. J. Sound Vib.
**218**(3), 389–403 (1998)MathSciNetCrossRefGoogle Scholar - 50.Perlikowski, P., Yanchuk, S., Wolfrum, M., Stefanski, A., Mosiolek, P., Kapitaniak, T.: Routes to complex dynamics in a ring of unidirectionally coupled systems. Chaos Interdiscip. J. Nonlinear Sci.
**20**(1), 013111 (2010)MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.