Fractional order new complex hyper-chaotic system with parameter attractor
阅读说明：本技术 一种具有参数吸引子的分数阶新复超混沌系统 (Fractional order new complex hyper-chaotic system with parameter attractor ) 是由 张芳芳 李正峰 孙凯 张雪 纪鹏 于 2021-07-12 设计创作，主要内容包括：本发明涉及混沌系统领域,具体地涉及一种具有参数吸引子的分数阶新复超混沌系统,建立的具有参数吸引子的分数阶新复超混沌系统并给出了其建立过程,本发明随着某个参数的不断增大或减小,会经历多个混沌和周期状态,然后直到发散,系统的吸引子也会呈现出不同的形式。此外,本发明的新复超混沌系统不仅具有蝴蝶吸引子形式,而且还具有类蝴蝶吸引子。随着参数的变化,系统至少经历两种吸引子形式,称之为参数吸引子；本发明的新复超混沌系统具有更高的维数和更多的参数,可以提高保密通信的安全性。(The invention relates to the field of chaotic systems, in particular to a fractional order new hyperchaotic system with a parameter attractor, which is established and gives an establishing process. In addition, the novel complex hyper-chaotic system not only has a butterfly attractor form, but also has a butterfly-like attractor. As the parameter changes, the system undergoes at least two attractor forms, referred to as parametric attractors; the new complex hyper-chaotic system has higher dimension and more parameters, and can improve the security of secret communication.)
1.A fractional order new complex hyper-chaotic system with a parameter attractor is characterized in that: the chaotic system is in the form of:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l ═ 1, 2, 3, 4) order; x is the number of1＝Is a complex variable, x3，x4As a real variable, a, b, c, d are system parameters, and when the system parameter a is 40, b is 25.5, c is 10, d is 12, the order α ∈ [0.9,1.01 ∈ []When the system is in a chaotic state.
2. The fractional order new hyper-chaotic system with parametric attractor of claim 1, wherein: the chaotic system is established by the following steps:
(1) definition 1.Caputo definition is also called right definition, and its specific expression is:
whereinα＝[m]+1，[m]Is the integer part of m, Γ (·) is a gamma function andreferred to as the alpha order differential operator; by usingTo representAnd alpha is more than 0 and less than 1;
(2) constructing a fractional order new complex hyper-chaotic system with a parameter attractor, wherein the form is as follows:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l ═ 1, 2, 3, 4) order;is a complex variable, x3，x4Is a real variable, ab, c, d are system parameters;
according to the fractional order differential linear algorithm, the method can be obtained
The system (4) can be transformed into the following real representation:
(3) fractional order attractor phase diagram:
selecting the initial value of the system (6) asx3＝1，x41, fixed parameters a 40, b 25.5, c 10, and d 12; the order of the selected systems (4) is the same and alphal0.95(l ═ 1, 2, 3, 4); under the order and the parameters, an attractor phase diagram of the system presents a butterfly attractor;
(4) influence of system order alpha on butterfly attractors:
when α <0.83, the system (6) diverges, so the control order α increases from 0.83;
when the alpha belongs to the interval of 0.83 and 0.9, the system (6) gradually converges to a stable point;
when alpha is 0.9, the system evolves from a fixed point to a chaotic state;
when alpha belongs to [0.9,1.01], the system is in a chaotic state;
when alpha is 1.02, the system (6) enters a periodic state;
when α >1.11, the system (6) diverges.
3. The fractional order new hyper-chaotic system with parametric attractor, as claimed in claim 1 or 2, wherein: when the order α is 1.01, the system will present a butterfly attractor shape.
4. The fractional order new hyper-chaotic system with parametric attractor of claim 3, wherein: keeping the parameters b, c and d unchanged, and controlling the parameter a to change, the system (6) gradually evolves from a limit ring state a of 37 to a butterfly-shaped attractor shape a of 39.5, then a central ring of the butterfly-shaped attractor continuously diffuses a of 40, and finally evolves to a butterfly-shaped attractor shape a of 42.
5. The fractional order new hyper-chaotic system with parametric attractor of claim 3, wherein: keeping the parameters a, c and d unchanged, controlling the parameter b to change, gradually evolving the system (6) from the butterfly attractor shape b being 23.5 to the butterfly attractor b being 24, then continuously contracting the center loop of the butterfly attractor b being 25.5, and finally evolving to the limit loop b being 30.9.
6. The fractional order new hyper-chaotic system with parametric attractor of claim 3, wherein: keeping the parameters a, b and d unchanged, controlling the parameter c to change, selecting the parameter c with the same value as the integer order in the section to observe the evolution of the system, and gradually evolving the system (6) from a butterfly attractor shape c equal to 5 into a butterfly attractor c equal to 9.3 and finally into divergence c equal to 2785.
7. The fractional order new hyper-chaotic system with parametric attractor of claim 3, wherein: keeping the parameters a, b and c unchanged, controlling the parameter d to change, and gradually evolving the system (6) from the butterfly attractor shape d being 4.5 to the butterfly attractor d being 12 and finally evolving to the divergence d being 2785.
Technical Field
The invention relates to the field of chaotic systems, in particular to a fractional order new hyperchaotic system with a parameter attractor.
Background
In 1963, Lorenz obtained a complete set of third-order ordinary differential equations by simplifying a fluid convection model, and thus a Lorenz chaotic system was proposed. In 1999, a new chaotic system appeared, which is similar to the Lorenz chaotic system but has a different topology, and is called a Chen chaotic system. In 2002, the Lu chaotic system appears in transition from the Lorenz chaotic system to the Chen chaotic system. The chaotic systems are all real number domain chaotic systems. In 1982, Gibbon et al discovered a system similar to the real Lorenz system when studying the phenomenon of thermal convection and detuned laser in fluid mechanics, but with the main variables in the complex domain, called the complex Lorenz system. Then, in 2009, Gamal m.mahmoud et al proposed a complex lu chaotic system and a complex Chen chaotic system on the basis of the real lu chaotic system and the real Chen chaotic system, and realized synchronization of the two systems.
The hyper-chaotic system has higher dimensionality and more controllable parameters, and can enhance the safety of chaotic secret communication. Moreover, the hyperchaotic system can show more unique chaotic characteristics. Compared with the hyperchaotic system in the real number domain, the complex hyperchaotic system has a simpler system form but can represent a higher dimension. When applied to communications, bivariates may be used to increase the security of information transfer. Many scholars have proposed various complex hyperchaotic systems.
With the intensive research on complex chaotic systems, a fractional order chaotic system has become one of the current research hotspots. The fractional order complex chaotic system has the characteristics of the fractional order complex chaotic system and the complex chaotic system, so the dynamic behavior is richer, and the method has a wider application prospect in the field of secret communication.
There are few learners who study this problem in the structure of a complex chaotic system. Therefore, it is necessary to provide a new fractional order complex hyper-chaotic system.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a fractional order new hyperchaotic system with parameter attractors, wherein at least two chaotic attractors appear when the parameters are changed.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a fractional order new complex hyper-chaotic system with a parameter attractor is in the form of:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l ═ 1, 2, 3, 4) order; is a complex variable, x3,x4As a real variable, a, b, c, d are system parameters, and when the system parameter a is 40, b is 25.5, c is 10, d is 12, the order α ∈ [0.9,1.01 ∈ []When the system is in a chaotic state.
The chaotic system is established by the following steps:
(1) definition 1.Caputo definition is also called right definition, and its specific expression is:
wherein α ═ m]+1，[m]Is the integer part of m, Γ (·) is a gamma function andreferred to as the alpha order differential operator. In this context, useTo representAnd alpha is more than 0 and less than 1.
(2) Constructing a fractional order new complex hyper-chaotic system with a parameter attractor, wherein the form is as follows:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l is 1, 2, 3, 4).Is a complex variable, x3，x4Is a real variable, and a, b, c and d are system parameters.
According to the fractional order differential linear algorithm, the method can be obtained
The system (4) can be transformed into the following real representation:
(3) fractional order attractor phase diagram:
selecting the initial value of the system (6) asx3＝1，x4The fixed parameters a, b, c, and d are 1, 40, 25.5, 10, and 12, respectively. The order of the selected system (4) is the same and al0.95(l ═ 1, 2, 3, 4); at this order and parameter, the attractor phase diagram of the system presents a butterfly attractor.
(4) Influence of system order alpha on butterfly attractors:
when α <0.83, the system (6) diverges, so the control order α increases from 0.83.
When the alpha belongs to the interval of 0.83 and 0.9, the system (6) gradually converges to a stable point;
when alpha is 0.9, the system evolves from a fixed point to a chaotic state;
when alpha belongs to [0.9,1.01], the system is in a chaotic state;
when alpha is 1.02, the system (6) enters a periodic state;
when α >1.11, the system (6) diverges.
When the order α is 1.01, the system will present a butterfly attractor shape.
Keeping the parameters b, c and d unchanged, and controlling the parameter a to change, the system (6) gradually evolves from a limit ring state a of 37 to a butterfly-shaped attractor shape a of 39.5, then a central ring of the butterfly-shaped attractor continuously diffuses a of 40, and finally evolves to a butterfly-shaped attractor shape a of 42.
Keeping the parameters a, c and d unchanged, controlling the parameter b to change, gradually evolving the system (6) from the butterfly attractor shape b being 23.5 to the butterfly attractor b being 24, then continuously contracting the center loop of the butterfly attractor b being 25.5, and finally evolving to the limit loop b being 30.9.
Keeping the parameters a, b and d unchanged, controlling the parameter c to change, selecting the parameter c with the same value as the integer order in the section to observe the evolution of the system, and gradually evolving the system (6) from a butterfly attractor shape c equal to 5 into a butterfly attractor c equal to 9.3 and finally into divergence c equal to 2785.
Keeping the parameters a, b and c unchanged, controlling the parameter d to change, and gradually evolving the system (6) from the butterfly attractor shape d being 4.5 to the butterfly attractor d being 12 and finally evolving to the divergence d being 2785.
The fractional order of the hyperchaotic complex system at least generates two attractors along with the change of parameters, namely, the parameter attractor exists.
The invention has the technical effects that:
compared with the prior art, the fractional order new hyperchaotic system with the parameter attractor has the advantages that as a certain parameter is continuously increased or reduced, the fractional order new hyperchaotic system can experience a plurality of chaotic and periodic states, and then the attractor of the system can also present different forms until divergence. In addition, the novel complex hyper-chaotic system not only has a butterfly attractor form, but also has a butterfly-like attractor. As the parameter changes, the system undergoes at least two attractor forms, referred to as parametric attractors; the new complex hyper-chaotic system has higher dimension and more parameters, and can improve the security of secret communication.
Drawings
FIG. 1 is a phase diagram of a fractional order attractor according to the present invention;
FIG. 2 is a graph of attractors as a function of order for the system (5) of the present invention;
FIG. 3 is a phase diagram of the attractor as a function of parameter a in accordance with the present invention;
FIG. 4 is an attractor phase diagram as a function of parameter b;
FIG. 5 is an attractor phase diagram as a function of parameter c for the present invention;
FIG. 6 is an attractor phase diagram as a function of parameter d according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the drawings of the specification.
Example 1:
to date, fractional calculus has mainly three definitions: Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. Since the traditional initial conditions and constant expressions are included in the Caputo definition, and take into account the engineering applicability of Caputo calculus, in the present invention, the Caputo definition is used.
The fractional order new complex hyper-chaotic system with the parameter attractor related by the embodiment has the following form:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l is 1, 2, 3, 4).Is a complex variable, x3，x4As a real variable, a, b, c, d are system parameters, and when the system parameter a is 40, b is 25.5, c is 10, d is 12, the order α ∈ [0.9,1.01 ∈ []When the system is in a chaotic state; the chaotic system is established by the following steps:
1. definition 1.Caputo definition is also called right definition, and its specific expression is:
wherein α ═ m]+1，[m]Is the integer part of m, Γ (·) is a gamma function andreferred to as the alpha order differential operator. In this context, useTo representAnd alpha is more than 0 and less than 1.
2. Constructing a fractional order new complex hyper-chaotic system with a parameter attractor, wherein the form is as follows:
whereinIs alphalOrder Caputo operator, alphalIs a corresponding state variable xl(l is 1, 2, 3, 4).Is a complex variable，x3，x4Is a real variable, and a, b, c and d are system parameters.
According to the fractional order differential linear algorithm, the method can be obtained
The system (4) can be transformed into the following real representation:
3. fractional order attractor phase diagram
Selecting the initial value of the system (6) asx3＝1,x4The fixed parameters a, b, c, and d are 1, 40, 25.5, 10, and 12, respectively. The order of the selected system (4) is the same and al0.95(l is 1, 2, 3, 4). On the basis of the Caputo definition, the system (4) is simulated by Matlab to obtain an attractor phase diagram of the system, which is shown in FIG. 1.
4. Influence of system order alpha on butterfly attractors
When α <0.83, the system (6) diverges, so the control order α increases from 0.83.
When the alpha belongs to the interval of 0.83 and 0.9, the system (6) gradually converges to a stable point;
when alpha is 0.9, the system evolves from a fixed point to a chaotic state;
when alpha belongs to [0.9,1.01], the system is in a chaotic state;
when alpha is 1.02, the system (6) enters a periodic state
When α >1.11, the system (6) diverges. Fig. 2 shows the system phase diagram when the order α takes different values.
5. Influence of system parameters on fractional order butterfly attractors
When the order alpha is 1.01, the system can present a butterfly attractor shape, the order alpha is 1.01, parameters a, b, c and d are respectively controlled to change, and the attractor change of the system is further observed.
(1) Influence of parameter a
And keeping the parameters b, c and d unchanged, controlling the parameter a to change, and observing the evolution of the system by selecting the parameter a with the value same as that in section 2.1 in the section. As shown in fig. 3, it can be seen that the system (6) gradually evolves from a limit loop state (a ═ 37) to a butterfly attractor shape (a ═ 39.5), then the central loop of the butterfly attractor continuously diffuses (a ═ 40), and finally evolves to a butterfly attractor shape (a ═ 42).
(2) Influence of parameter b
And keeping the parameters a, c and d unchanged, controlling the parameter b to change, and observing the evolution of the system by selecting the parameter b with the same value as the integer order in the section. As shown in fig. 4, it can be seen that the system (6) gradually evolves from a butterfly attractor shape (b ═ 23.5) to a butterfly attractor (b ═ 24), then the central loop of the butterfly attractor continuously shrinks (b ═ 25.5), and finally evolves to a limit loop (b ═ 30.9).
(3) Influence of the parameter c
The parameters a, b and d are kept unchanged, the parameter c is controlled to change, and in the section, the value of the parameter c is selected to be the same as the value of the integer order, so that the evolution of the system is observed. As shown in fig. 5, it can be seen that the system (6) evolves gradually from a butterfly attractor shape (c-5) to a butterfly attractor (c-9.3) and finally to divergence (c-2785).
(4) Influence of parameter d
The parameters a, b and c are kept unchanged, the parameter d is controlled to change, and in the section, the value of the parameter d is selected to be the same as the value of the integer order, so that the evolution of the system is observed. As shown in fig. 6, it can be seen that the system (6) gradually evolves from a butterfly attractor shape (d ═ 4.5) to a butterfly attractor (d ═ 12) and finally to divergence (d ═ 2785).
By observing the change of the shape of the attractor of the fractional order new hyper-chaotic system along with the parameters, the phenomenon that the fractional order has the parameter attractor can be seen.
The fractional order of the new hyperchaotic complex system generates at least two attractors along with the change of parameters, namely, a parameter attractor exists.
Based on the analysis and observation of the new hyperchaotic complex system fractional order evolution process, a comparison table of the influence of the parameter change on the fractional order system under four conditions is obtained, as shown in tables 1-4. In tables 1 to 4Fks.i，(k＝a，b，c，d)(i＝1，2，3，..，n)Refers to attractors with different shapes generated by the change of parameters a, b, c and d.
Table 1.a changes, b 2.5, c 10, d 12, (i (f)as.i，(i＝1，2，3)I (i 1, 2, 3.., n)) of the attractor.
Table 2.b changes, a-40, c-10, d-12, (i (f)bs.i，(i＝1，2)I (i 1, 2, 3.., n)) of the attractor.
Table 3.c changes, a-40, b-25.5, d-12, (i (f)cs.i，(i＝1，2，3)I (i 1, 2, 3.., n)) of the attractor.
Table 4.d changes, a-40, b-25.5, c-10, (i (f)ds.i，(i＝1，2，3，4)I (i 1, 2, 3.., n)) of the attractor.
It can be seen that the fractional order of the new hyperchaotic complex system produces at least two attractors with the change of the parameters, namely, the parameter attractor exists.
The above embodiments are only specific examples of the present invention, and the protection scope of the present invention includes but is not limited to the product forms and styles of the above embodiments, and any suitable changes or modifications made by those skilled in the art according to the claims of the present invention shall fall within the protection scope of the present invention.
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