JPH096406A - Chaos control method and its device - Google Patents

Abstract

PURPOSE: To provide a chaos control method and its device in which an unstable point of a chaos dynamic system generating a chaos is made stable. CONSTITUTION: An unstable point of a chaos dynamic system is obtained (S1), a Jacobian with respect to a system provided with a feedback for control is obtained (S2), the range of parameters of the feedback is obtained so that all absolute values of specific values are smaller than the unity (S3), and the unstable point is made stable by applying feedback to the chaos dynamic system by using the feedback parameter and the unstable point.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a chaos control method and apparatus for stabilizing an unstable point in a chaotic dynamic system, and can be used in the industrial field of control and the like.

[0002]

2. Description of the Related Art A conventional chaotic control method for stabilizing an unstable point in a chaotic dynamical system (a non-linear equation of a circuit or a phenomenon, a model, etc.) is different from the case of adding a feedback term depending on how it is added. , There are two cases of multiplicative multiplication of the parameters, but in both cases, the system may diverge due to the feedback amount. A conventional chaos control method is shown in FIG. FIG. 3 is a flowchart for explaining a conventional chaos control method, and (S11) to (S14) show respective steps.

First, the unstable point of the chaotic dynamical system to be controlled is obtained (S11). Next, the Jacobian for the system including the feedback for control is calculated (S12). Then, the range of the feedback parameter is calculated so that the absolute values of the Jacobian eigenvalues are all smaller than 1 (S13). Then, the instability obtained in step (S11) and the feedback parameter obtained in step (S13) are used to feed back to the target chaotic dynamical system to stabilize the unstable point (S14).

[0004]

In the above-described conventional chaos control method, since feedback is simply performed using the unstable point and the feedback parameter as described above,
There was a problem that the system diverged.

The present invention has been made in order to solve the above-mentioned problems of the prior art, and its purpose is to provide feedback such that the system does not diverge with respect to a target chaotic dynamical system. The object of the present invention is to provide a chaos control method and apparatus capable of stabilizing an unstable point and controlling a chaotic state to the stable point.

[0006]

A chaos control method according to the present invention obtains an instability point of the target chaotic dynamical system, obtains a Jacobian for a system including feedback for control, and obtains an eigenvalue thereof. The range of the feedback parameters is calculated so that all the absolute values of are smaller than 1, and the feedback parameters and the instability points thus obtained are used to perform feedback so that the system does not diverge to the target chaotic dynamical system. As a result, the unstable point is stabilized.

Further, the chaos control device according to the present invention includes an unstable point calculation circuit for obtaining an unstable point of a chaotic dynamical system circuit (a circuit which becomes a chaotic dynamical system, such as an electric circuit or an optical circuit) to be controlled, A Jacobian calculation circuit for obtaining a Jacobian of the chaotic dynamical system circuit including a feedback circuit, a feedback parameter calculation circuit for obtaining a range of feedback parameters so that the absolute values of the Jacobian eigenvalues are all smaller than 1, and the chaotic dynamical system A feedback circuit for stabilizing the unstable point of the chaotic dynamical system by giving a feedback for stabilizing the unstable point such that the system does not diverge.

[0008]

In the chaos control method and apparatus of the present invention, feedback parameters and instability points are used for the chaotic dynamical system to be controlled, and feedback is performed so that the system does not diverge to the target chaotic dynamical system. As a result, the unstable point is stabilized and the chaotic state is controlled to the stable point.

[0009]

EXAMPLE An example of this example will be described below.

FIG. 1 shows a flow chart of an embodiment of the chaos control method of the present invention. Further, (S1) to (S4) show each step.

Two cases will be considered, depending on how the feedback term is added, by way of example: additive addition and multiplicative multiplication of parameters.

First, consider the case of additive addition. Let the chaotic dynamical system be a discrete dynamical system x (t + 1) = f (x (t)). The procedure is almost the same for continuous dynamical systems. The feedback term for controlling the function F (t) with the following saturation characteristics
As to prevent the divergence of the system.

[0013]

[Equation 1] However, p is a feedback parameter and x _d (t) is an unstable orbit to be stabilized. For example, the unstable trajectory x _{d in} the case of cycle 1 is obtained by the following formula (S1).

[0014]

[Equation 2] Next, the Jacobian for the system including the feedback for control is calculated (S2), and the range of the feedback parameter is calculated so that the absolute values of the eigenvalues are all smaller than 1. The Jacobian J _control is given by the following equation.

[0015]

(Equation 3) The eigenvalue of this Jacobian is obtained by numerical calculation, and the feedback parameter p is determined so that the absolute values of the eigenvalue are all smaller than 1 (S3).

The unstable point can be stabilized by the unstable trajectory x _d and the feedback parameter p, and the chaotic state can be controlled to the stable point (S4).

Further, with respect to the multi-period point, the same calculation is performed, and the feedback parameter p at each multi-period point may be determined.

As a next example, consider the case where the parameters are multiplied in a multiplicative manner. A chaotic dynamical system is a discrete dynamical system x (t + 1) = f (x
(t); μ). μ is a parameter of the dynamic system. The procedure is almost the same for continuous dynamical systems. The feedback term for controlling the function F with the following saturation characteristics
(t).

[0019]

(Equation 4) However, p is a feedback parameter and x _d (t) is an unstable orbit to be stabilized (S1). Similarly, the unstable trajectory x _{d in} the case of the period 1 is obtained by the following formula.

[0020]

(Equation 5) Next, the Jacobian J _control for the system including the feedback for _control is calculated (S2), the range of the feedback parameter p is calculated so that the absolute values of its eigenvalues are all smaller than 1, and the value of p is calculated. Determine (S3).
Similarly, the unstable point can be stabilized by the unstable orbit x _d and the feedback parameter p, and the chaotic state can be controlled to the stable point (S
4).

Further, regarding the multi-period point, the same calculation is performed, and the feedback parameter p at each multi-period point may be determined.

FIG. 2 is a block diagram showing the configuration of an embodiment of the chaos control device of the present invention.

In FIG. 2, reference numeral 1 denotes a chaotic dynamic system circuit to be controlled, which is a non-linear dynamic type circuit capable of generating chaos. 2 is an unstable point calculation circuit, which is a chaotic dynamics system circuit 1
This is a circuit for calculating the unstable point of. Reference numeral 3 denotes a Jacobian calculation circuit, which is a circuit for obtaining the Jacobian of the system of the chaotic dynamical system circuit 1 including the feedback circuit. Reference numeral 4 is a feedback parameter calculation circuit, which is a circuit for obtaining the range of feedback parameters so that all the absolute values of the Jacobian eigenvalues are smaller than 1. Reference numeral 5 denotes a feedback circuit, which is a circuit for giving feedback to the chaotic dynamics system circuit 1 for stabilizing the unstable point so that the system does not diverge.

Since its operation is the same as that described with reference to the flow chart of FIG. 1, its details are omitted.

[0025]

As described above, the chaos control method and apparatus of the present invention uses the feedback parameter calculated from the unstable point and the Jacobian for the chaotic dynamic system so that the system does not diverge to the target chaotic dynamic system. It is possible to stabilize the unstable point and control the chaotic state to the stable point by performing such feedback.
Therefore, when the current change is chaotic in an electric circuit, an optical circuit, or when only the frequency component of the laser oscillation system is chaotically fluctuating, the present invention can be applied to obtain a stable output. If applied, stable control can be performed.

[Brief description of drawings]

FIG. 1 is a flowchart of an embodiment of a chaos control method of the present invention.

FIG. 2 is a block diagram showing the configuration of an embodiment of the chaos control device of the present invention.

FIG. 3 is a flow chart diagram for explaining a conventional chaos control method.

[Explanation of symbols]

 1 Chaotic dynamics circuit 2 Instability calculation circuit 3 Jacobian calculation circuit 4 Feedback parameter calculation circuit 5 Feedback circuit

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Noboru Sonehara 1-1-6 Uchisaiwaicho, Chiyoda-ku, Tokyo Nihon Telegraph and Telephone Corporation

Claims (2)

[Claims] 1. A chaotic control method for stabilizing an instability point of a chaotic dynamical system, wherein an instability point of the chaotic dynamical system of interest is obtained, and a Jacobian for a system including feedback for control is calculated. Then, the range of the feedback parameter is calculated so that the absolute values of the eigenvalues are all smaller than 1, and the calculated feedback parameter and the unstable point are used so that the system does not diverge to the target chaotic dynamical system. A chaos control method characterized in that the unstable point is stabilized by performing various kinds of feedback. 2. An unstable point calculating circuit for obtaining an unstable point of a chaotic dynamical system circuit to be controlled, a Jacobian calculating circuit for obtaining a Jacobian of the chaotic dynamical system circuit including a feedback circuit, and an absolute value of an eigenvalue of the Jacobian. A feedback parameter calculation circuit for obtaining a range of feedback parameters so that all are smaller than 1, and a feedback for stabilizing an unstable point such that the system does not diverge to the chaotic dynamical system, And a feedback circuit for stabilizing the unstable point of the chaos control device.