A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.

A chaotic system is a highly sensitive nonlinear system. The main characteristic of chaos is the sensibility to the initial conditions. This sensibility is sometimes known as “butterfly effect” in honor of Lorenz [

The Lorenz system has been extensively employed: cyphers [

In order to increase its degree of complexity, several works have tried to modify the Lorenz dynamics. Some of them try to increase the number of fixed points in the system [

Among all these previous works, a very important group of articles are those focused on the creation of Lorenz-like “hyperchaotic systems.” Traditionally, hyperchaos was defined [

Therefore, the aim of this article is to describe a new hyperchaotic system, being able to generate highly complex and novel structures. For that, we seek to strengthen at least one direction of expansion and reduce the rate of contraction (but they should remain dominant in order to preserve the system as dissipative).

The rest of the paper is organized as follows. In the next section, a study about the redundant data in the components of Lorenz’s dynamics is presented. In Section

It has been proven in many articles that Lorenz system can be self-synchronized [

Consider two Lorenz systems (

A new function, called error function, is defined as follows:

Thus, both coupled Lorenz systems get synchronized in amplitude and phase if

The first differential equation is decoupled, so it may be solved directly (

Then, the resulting differential system (

A Lyapunov function satisfying two conditions is proposed as follows:

Theorem

The information about the system in any of the Lorenz dynamic components allows any intruder system (

Two first differential equations (

Besides, both error functions for the temporal evolution of the systems and for the parameters values are also defined as follows:

If it is satisfied that

As it is said previously, the first equation is decoupled and it can be proven that

A Lyapunov function is proposed (

Then, Lyapunov’s theorem about the asymptotic stability guarantees that

Thus, any intruder system may synchronize the temporal evolution of the transmitter, deduct the values of the parameters and, in conclusion, break the cryptosystem. For that, the same synchronization signal employed in the receiver is enough.

In conclusion, Lorenz-based cryptosystems may be broken in a pretty easy way due to the low degree of complexity of that dynamics. Moreover, Orúe et al. [

In order to get complex hyperchaos, we have considered the Lorenz system (

First of all, it must be noted that the proposed system still preserves the symmetry about

First return map for (a) Lorenz system

On the other hand, the proposed system presents three different fixed points (

As can be seen,

Other important requirements for the novel system are as follows. (i) We maintain the two nonlinear terms of the Lorenz system but we introduce one new divergence in the second equation of system (

Expressing the dynamics as a matrix, we obtain the following:

So, the dynamics parameters must verify the following condition:

As can be seen, the described dynamics depends on four parameters. However, in order to analyze the new system, it is important to choose a parameter configuration which not only maintains the system’s complexity but also simplifies its analysis. Therefore, we are taking fixed values for some system parameters. We fix the degree of divergence which controls the parameter

For

According to these arguments and in order to study the maximum number of different representative structures and topologies, three different analyses are going to be performed:

System evolution when varying the parameter

System evolution when varying the parameter

System evolution when varying the parameter

For all the three analyses described above, the eigenvalues associated with the matrix of the linearized system around the equilibrium points have no analytical expression depending on the dynamics parameters. Thus, to obtain their values and study the stability of the fixed points, possible bifurcations, and so forth, we made a numerical calculation. We take advantage in considering the symmetry of the system and we know, in general, that the linearized systems around the points

Local stability study.

Analysis | Parameters | Eigenvalues | Interpretation | Eigenvalues | Interpretation |
---|---|---|---|---|---|

System evolution when varying the parameter | | | Saddle point order 1 | | Saddle spiral order 2 |

| | Saddle point order 1 | | Saddle spiral order 2 | |

| | Saddle spiral order 1 | | Saddle spiral order 2 | |

| Hopf bifurcation | ||||

| | Saddle spiral order 1 | | Spiral node | |

| |||||

System evolution when varying the parameter | | | Saddle spiral order 1 | | Saddle spiral order 2 |

| |||||

System evolution when varying the parameter | | | Saddle spiral order 1 | | Saddle spiral order 2 |

| Hopf bifurcation | ||||

| | Saddle spiral order 1 | | Spiral node | |

| | Saddle point order 1 | | Spiral node |

From the analysis of the results shown in Table

Taking

On the other hand, with

Various methods to determine the supercritical character of Hopf bifurcations have been described in the research literature. In particular, a Hopf bifurcation is considered subcritical if the maximum Lyapunov exponent in the bifurcation point is positive and supercritical in the opposite case.

Numerical algorithms allow determining the Lyapunov spectrum of the two cases of interest. Results are shown in Table

Lyapunov spectrum at the bifurcation points.

Case | Lyapunov spectrum |
---|---|

| |

| |

In the proposed system, the complexity of the trajectories tends to increase for small values of parameter

Regular structures may be found in any of the three studies proposed. In particular, two different regular types of trajectories are generated by the proposed system. On one hand, common limit cycles are generated in the neighborhoods of the Hopf bifurcations identified in the previous section. On the other hand, more complex cycles are generated for different values of the parameter

In the next subsections, both cases are studied.

Limit cycles (Figure

Projection over the subspace

In both cases, two different limit cycles coexist at the same time (as can be seen in Figure

Coexisting limit cycles in the proposed system. (a)

This coexistence is maintained while the system remains in the regular region and disappears when chaotic trajectories appear (as these trajectories are developed around both fixed points). This fact is clearly shown in Figure

Bifurcation diagram using

Finally, the existence of a Hopf bifurcation must be proven. In Figure

The evolution of the limit cycles with the parameter

If the system flow evolution when varying the parameter

Bifurcation diagram using

Several and interesting bifurcations may be identified in these diagrams. For example, at

In some of the shown areas, limit cycles appear (e.g., in the range

Complex limit cycle for (a)

On the other hand, although the couple

Coexisting limit cycles for

In the proposed system, the weakly chaotic structures maintain certain similarity with the classic Lorenz attractor (see Figure

Lorenz-like chaotic attractor in the proposed system

Bifurcation diagram using

Other interesting structures in the phase space, together with the associated first return map, can be seen in Figure

Chaotic attractors of the

As we have said previously and as can be seen in the bifurcation diagram of Figure

Table

Lyapunov exponents study.

Topology | Lyapunov exponents | Kaplan-Yorke dimension |
---|---|---|

| | 2.12 |

| | 2.13 |

| | 2.47 |

In particular, Figure

Chaotic tridimensional attractors generated by the proposed system. Initial conditions:

As can be seen and as was said in Section

In this section, different complex chaotic structures are located and analyzed. As we will see below, in the proposed system, the strengthening of the expansion dimensions has motivated the appearance of highly complex chaotic trajectories, sometimes coexisting with hyperchaotic ones. Other complex structures (such as the chaotic-transient solutions) and unbounded solutions, when the parameter

These structures are reviewed in the following subsections.

As we have commented in the previous subsection, hyperchaotic and unbounded solutions coexist in most of the cases. However, another type of solutions which may be confused with hyperchaotic structures can exist. In some cases, the proposed system generates trajectories which seem to be hyperchaotic at first. However, as time elapses, the components enter in a coherent state which does not correspond with hyperchaos. This situation, for example, appears for the couple of parameters

Figure

Projection over the subspace

In fact,

Temporal evolution in the chaotic-transient solutions.

Figure

First return map for the chaotic-transient solution. (a)

Although, probably, different hyperchaotic structures could be generated by the proposed system using different values of the control parameters, in this section, we will focus the study on the behavior in the range

The first hyperchaotic structure may be found at

Different projections of the hyperchaotic attractor. Initial conditions:

A basic proof of the presence of hyperchaos is obtained by means of the Lyapunov spectrum. Table

Lyapunov exponents study for the hyperchaotic structure.

Exponent | Value | Kaplan-Yorke dimension |
---|---|---|

| 12.37 | 3.83 |

| 0 | |

| −0.036 | |

| −14.71 |

Using the ordered Lyapunov spectrum

Also, we can observe that

For

For

Our case belongs to the more complex chaotic behavior indicated in (b).

One of the problems when working with dynamical systems with a high level of complexity is the flow instability depending on the control parameters in the region of hyperchaos and transient chaos. This high level of complexity appears for low values of

In the case of the result shown in Table

Evolution of the Lyapunov exponents values versus the integration time.

Finally, since the four-dimensional attractors cannot be represented in only one graphic, Poincaré sections are a traditional way of proving the complexity level and the presence of chaos. Although the Lyapunov exponents and the previous results prove the hyperchaotic behavior of the analyzed structure, Poincaré sections in Figure

Poincaré sections of the system for different

As can be seen in Figure

Bidimensional

In particular, this new topology exhibits chaotic behavior, with a great complexity, and develops an attractor (see Figure

Lyapunov exponents study.

Exponent | Value | Kaplan-Yorke dimension |
---|---|---|

| 1.2092 | 2.52 |

| 0 | |

| −2.3517 | |

| −2.8622 |

Tridimensional projections of the highly complex chaotic attractor coexisting with the hyperchaotic one. Initial conditions:

In this article, a new hyperchaotic four-dimensional Lorenz-based system, especially designed to improve the complexity of traditional Lorenz dynamics, is numerically analyzed. First, a proof about the weakness of the Lorenz system, due to the amount of redundant information present in its components, is provided. Regular, chaotic, and hyperchaotic structures are located and analyzed by varying two parameters of the system. Analyses about the symmetry, stability, and bifurcation are also provided. Two Hopf bifurcations and other typical bifurcations such as period doubling bifurcations and a tangent bifurcation were observed when

The authors declare that there are no competing interests regarding the publication of this paper.

One of the authors, Borja Bordel, has received funding from the Ministry of Education through the FPU Program (Grant no. FPU15/03977), the Ministry of Economy and Competitiveness through SEMOLA project (TEC2015-68284-R) and from the Autonomous Region of Madrid through MOSI-AGIL-CM project (Grant P2013/ICE-3019, cofunded by EU Structural Funds FSE and FEDER). The authors are grateful for discussions with Professor Vicente Alcober.