A Note on the Exponents of Lyapunov for a System with Unknown Dynamic

Objective: To determine the exponents of Lyapunov for a system with unknown dynamics.


Introduction
Dynamic systems can be described by a set of ordinary differential equations, which when going into detail in terms of their modeling increases their complexity, becoming non-linear. The analysis of linear dynamic systems in phase space has reduced behaviors (fixed points, periodic or divergent solutions); in terms of non-linear systems they contain a variety of scenarios, stationary or convergent solutions (fixed points), periodic solutions (limit cycles), quasi-periodic (attractor bull) or chaotic solutions (strange attractors).
The chaos has been widely studied by mathematicians and engineers since its discovery by Edward Lorenz in 1970s. The main characteristic of these systems is the extreme sensitivity to the initial conditions, that is, for nearby trajectories initially their distance in the state space diagram will increase exponentially, being quantified by the exponents of Lyapunov 1 .
If you want to determine the exponents of Lyapunov for a system with unknown dynamics, you must reconstruct the phase space that captures the original attractor of the system. The most familiar method for the reconstruction of phase space from a scalar time series is based on the determination of a delay time and an embedding dimension 2 .

Phase Space
When observing a real multivariable dynamic system, it is normal that all it's variables cannot be measured simultaneously, in the worst case there is only one measurement (time series).
where T is the transpose. The proper choice of delay time and embedment dimension is very important for the correct reconstruction of the attractor. There are methods to determine the delay time, such as the false neighboring neighbors. On the other hand, the dimension of embedding of a n-dimensional system is obtained according to the formulation of Takens 3 .

Lyapunov Exponents
For an n-dimensional system described by the i-th exponent of Lyapunov is defined i λ depending on the variations of the trajectories from a starting point 0 x in 0 t , these variations can be seen as the degeneration of the radius of a hyper sphere 0 (t) δ in a hyper-ellipsoid of radio i (t) δ The classic procedure for the estimation of the exponents of Lyapunov was proposed by Wolf in 1985 4 , which consists of evaluating the local divergences from the application of the tangent plane associated with the system of state equations on n orthogonal vectors Initially defined by the identity matrix n I The variational expressions give rise to the tangent space, defined as: Where J(x,t) represents the Jacobian of F(x,t). The system of equation of state together with the variational expres- for a time T and thus obtain the divergent vectors transformed by the application of the tangent plane, that is, for the first exponent and the superscript represents the current iteration of the calculation. By repeating this procedure of integration and normalization K times the i-th exponent of Lyapunov can be written in the following way: In each iteration made the direction of the vectors it must be aligned with the expansion direction of the system dynamics, for this reason the Gram-Schmidt method must be used, as shown below Once the exponents of Lyapunov have been calculated, the maximum number of them must be identified, since it is the one in charge of determining the presence of chaos or not in said dynamic system. For this, it is clear that if you have 0 max λ > the initially close trajectories in the phase space will tend to separate, that is, the system presents a high sensitivity to the initial conditions.

Application to Chua's Circuit
Since its discovery in 1983, the Chua circuit has served as the basis for the study of the chaos in electronic systems 5 . This circuit is composed of two capacitors (C1 and C2), an Pedro Pablo Cárdenas Alzate, José Gerardo Cardona Toro and Luz María Rojas Duque inductance (L), a resistance (R) and a non-linear resistance (Rn), in Figure 1 the canonical circuit can be observed. Solving the circuit by state variables, the system is described by differential equations: is the function that characterizes non-linear resistance, and is defined as follows?
Here, E is a parameter dependent on the polarization voltage of the circuit, when implemented with OPAMs. The mathematical model of the Chua circuit can be normalized according to the proposal by Matsumoto 5 , for greater ease in the simulation α, β being system parameters with typical values are α = 9, β = 14.286. Here, h(x) it is represented in the form: The values of the constants 0 m and 1 m are assumed for this analysis with -1/7 and 2/7 respectively. By integrating the system (9) using the Runge-Kutta45 algorithm, the time series is obtained (Figure 2).
Graphing the phase diagram between Vc1 and Vc2 shows that the resulting attractor is a limit cycle.
By varying the control parameter α=9 (Figure 3), a large change in the dynamics of the system is observed, becoming a chaotic system (strange attractor) as seen in the phase diagram of Figure 4.

Reconstruction of the Phase Space of Chua's Circuit
As mentioned, in various physical situations there is no measurement of all the variables of the system's state, therefore, it is necessary to reconstruct the dynamics of the system from the available measurements. Assuming that only the voltage measurement of capacitor one (Vc1) is available from the Chua circuit, is ( Figure 5): Taking as dimension of embedment m=3 and a delay time of =5s τ , the attractor of the system seen in Figure 6 is obtained.

Determination of Lyapunov Exponents for the Chua Circuit
For the calculation of the Lyapunov exponents 6 of the Chua circuit, we start from the system (9), from which the Jacobian matrix is identified: Which is the basis to start with the algorithm proposed by Wolf 4 , for the determination of the exponents of Lyapunov. The simulation (Figure 7) was performed with T=200, =9 α and initial conditions [0.15264 -0.02281 0.38127].     The values obtained are shown in Table 1. It is observed that the maximum exponent of Lyapunov 1 920399 0 max . λ = > , therefore the system will behave chaotic and its future states cannot be determined from the initial states. From the results of Figures 8-9, 4 a high dependence of the dynamics is observed with respect to the parameter α.

Conclusions
The characterization of the dynamics of any system is determined by exponents of Lyapunov, which, depending on their value, allow a qualitative analysis, establishing that dynamic is converging to a fixed point of the state space, or if it is periodic (limit cycle and bull) or simply if the future states cannot be determined from the initial states (strange chaotic-attractor system). Therefore, the constructive form of the circuit, for values between 7 and 11 there is a highly variable spectrum of the exponents of Lyapunov as it passes from one moment to operate stably under a limit cycle to be divergent. Finally, based on the knowledge of the dynamics of a system, a control law can be proposed to stabilize the system around the desired operating point that is why the phase space must be reconstructed when this dynamic is not known.

Acknowledgements
We would like to thank the referee for his valuable suggestions that improved the presentation of this paper and our gratitude to the Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia) and the group GEDNOL.