Study on a New Character of Chaotic States to a Nonlinear Measure Model : Pseudo Temperature

Chaos is ubiquitous and extremely complicated phenomenon in the decision of nonlinear systems. Bifurcations of periodic, quasi-periodic and chaotic oscillations are frequently observed in forced nonlinear circuits described by no autonomous differential equations with a periodic external forcing term. Qualitative analysis using the Poincare mapping is commonly applied in the study of bifurcation phenomena as well as in numerical analysis of periodic and non-periodic solutions. In the measurement of electronics, electrical engineering, when the measurement object is nonlinear, the results of measurement may occur as chaotic state. The study used the nonlinear circuit as the measure model, studies probability density of "Pseudo-Temperature", which can be used to represent the degree of chaotic states and provides a new way of quantitative analysis to chaotic states of the nonlinear measure system. The experiment results show Pseudo-Temperature is new characteristic parameters, it can measure wide-area information from the normal to the randomness in the chaotic state.


INTRODUCTION
Chaos is ubiquitous and extremely complicated phenomenon in the decision of nonlinear systems.Chaotic state is a steady-state response of nonlinear system which is always restricted to limited area and its track never overlaps with complicated behavior and random movements.In the measurement of electronics, electrical engineering, when the measurement object is nonlinear, the results of measurement may occur as chaotic state.
In order to give the background of chaotic states and to support the new concept of pseudo temperature, There are several algorithms had proposed for describing the chaotic phenomenon in the nonlinear circuit, such as Louodop et al. (2012), Chen and Wu (2012), Trzaska (2011), Cheng et al. (2008), Ning et al. (2006) and Kawakami (1992).
In this study, the purpose is to present some elementary mechanisms of the bifurcations and to discuss numerical methods for analyzing these nonlinear phenomena.As illustrated examples, numerical results of forced oscillatory circuits containing storable inductors are presented.We used the nonlinear circuit as the measure model.At the same time, we propose a new concept "Pseudo-Temperature" which can be used to represent when it comes into chaotic states and provides a new way of quantitative analysis to chaotic states of the nonlinear measure system.The experiment results show Pseudo-Temperature is new characteristic parameters and it's a Fig. 1: The RLC nonlinear circuit new way to measure wide-area information from the normal to the randomness in the chaotic state.

LITRATURE REVIEW
In Fig. 1, the RLC of nonlinear circuit connection in series can be regarded as a simplified model for some certain nonlinear systems, the circuit equation: where, C = The voltage-controlled variable capacitance diodes V c = The terminal voltage of the nonlinear capacitance Fig. 2: Phenomenon of bifurcation to chaotic state in the logistic system (2) Thus, Eq. ( 1) can be written as： Acting as a measurement of electric current of the excitation signal function V (t), also as other control parameters, ω, R, L, C0 and β function to study.For this type of circuit model, Lindsay and Testa, had observed the capacitor terminal voltage Vc (shown as the ordinate V y ) early in 1981 in the experiment.and the amplitude of the excitation signal V (t), V 0 (V 0 shown in the abscissa) in a relationship that the phenomenon of turning from the bifurcation of the twocycle state to the chaotic state (Fig. 2).Besides, we give the Logistic phase diagram of different values of µ with shown in Fig. 3. Using nonlinear circuit as the measure model, the paper studies the probability density and gives a new concept "Pseudo Temperature".
Further study into the impact on the Logistic system when the value is changing: Suppose the initial value is 0.6 and m changes from 2.6 to 4, then the Logistic iteration limit morphology diagram is shown in Fig. 2 (Thompson AND Stewart, 1986), with the value changes, the system dynamics form changes constantly and finally we will see chaotic state (Gonzalez and Woods, 1992;Jain, 1989).
Another common chaotic model is Lorenz system.American meteorologist Lorenz posed a strange attractor dynamics system in the performance of atmospheric science magazine in 1963.The chaotic system model can be described by the following differential equations: We use MATLAB mathematical software to solve the above differential equations and begin numerical simulations.At first, we need to establish the M-file Lorenz to define the script function and then we start the programming calls.
Fixed other parameters and set the initial values f0 and computation time t0 we can find the phenomenon by changing the parameters a so that the system gradually enters into the chaotic state of the process.The Fig. 4 gives us the circumstance of waveforms in the Lorenz model with different values.
We believe the chaotic state probability density and its Pseudo temperature are symbols that are measured objects in a chaotic state "chaos in the orderly", with big significance signing important regularity.And knowing the probability density, we will obtain the random process of the object, so that determines the chaotic state of the pseudo random process.It provides new characteristic parameters so that intelligent instruments can measure wide-area information from the normal to the randomness in the chaotic state

MATERIALS AND METHODS
Because the chaotic state is a pseudo-random state, we will take each pseudorandom state of chaotic state as spatial representation points, which we called "pseudo-particle" (Li and Yuan, 2008;Trzaska, 2011).Changes of the Representation Points in the phase space are equivalent to the pseudo-random variation of the chaotic state.Therefore, in order to describe the circuit model chaotic signals---pseudorandom changes of the chaotic state, we define state variables: ( 1) 1 2 ( ) ( ), ( ) ( ), ( ) ( ) n n q t q t q t q t q t q t Here we assume that the phase space is flat (q1, q2).Pseudorandom changes in the chaotic state are corresponding to the pseudo random motions of the representation point in the phase space and the dominated differential equations describe the point motions in the phase space has been proposed by Ning et al. (2006): According to the Fokker-Planck Eq. ( 3), (4): λ i , S ii , S ij are the coefficients of this equation, defined as: Following a series of calculations below, we can get the probability density of the chaotic state.First, from the formula (4), ( 5): Then from the formula (4), ( 5), ( 6) we can know: Base on F 1 = 0, F 2 = V 0 / L sin ωt According to the definition of formula ( 8), ( 9) and ( 10) we get this by calculating: Obviously, S 11 = 0, S 12 = 0 and we can get this from formula (8) by calculating: Take formula ( 12), ( 13) and ( 14) into formula (7), obtaining the nonlinear RLC Eq. ( 4), corresponding to the Fokker-Planck equation: For the steady state that ∂p/∂t =0, so the formula (15) turns into: Turn the formula ( 16) into variant as follows: (17) Then the formula (18) becomes: (18) Eq. ( 18) and the following equations can be solved: Similarly, comparing with formula (16): The solution of equation: (Zhang et al., 1998): Take Eq. ( 20) into ( 21), then we can get the solution of Eq. ( 18): Integration of the index above ‫‬ ܸ భ q 1 is the integration to the nonlinear potential energy Vc, it should be the electric potential energy stored in the nonlinear capacitor, however is the energy of motion, so the sum of the two is the total energy in the circuit, therefore we can get:

RESULTS
Eq. ( 23) is how we use the calculation method to obtain the description of Fig. 1, the probability density of chaotic state in the nonlinear circuit model.Obviously, it is similar in the shape with the distribution law of Boltzmann: We call this T * as "Pseudo Temperature".Obviously, the dimension remains true temperature.In this way, the formula (24) can be written as: (26)

CONCLUSION
The study used the nonlinear circuit as the measure model, studies probability density of "Pseudo-Temperature", which can be used to represent the degree of chaotic states and provides a new way of quantitative analysis to chaotic states of the nonlinear measure system.Some typical illustrative example above using chaotic circuit system has shown satisfactory performances to support the theory.
So, "Pseudo Temperature" is a new concept in the study.Pseudo temperature T * characterizes the random changes of the phase space-A kind of intensity describes the "Pseudo thermal motion''.At the same time, indirectly characterizes the nonlinear circuit model appears degree of pseudo-random chaotic state.Indirectly characterizes the degree of pseudo-random of chaotic state appears in the nonlinear circuit model.Similar research for other chaotic phenomena can be studied likewise.