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In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems’ solutions. The active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results.

In 1920, van der Pol invented the van der Pol oscillator [

Fractional calculus plays an essential role in modern science. It is a different and distinct method for dealing with nonlinear systems along with the integer order. Fractional order models are adequate for the description of dynamical systems rather than integer order models. We can recognize, describe, and know dynamic phenomena such as chaos, hyperchaos, synchronization, and some other aspects of fractional order models faster and more accurately than those of the integer order of nonlinear systems. At present, the application of fractional calculus in most scientific fields has attracted much attention. So, the fractional calculus on the dynamical system was essential and exciting, which had been investigated recently by many researchers [

Distributed order calculus has been investigated for the first time as the extinction of fractional order calculus by Caputo [

Synchronization of chaos has a critical part to play in dynamic systems. It has various applications in different fields [

System (

The principal aims of this paper described as follows. (1) The hyperchaotic generalized van der Pol method has been introduced with complex parameter distributed order (

The article is set out as follows. We are displaying some critical preliminaries in Section

The following section includes some definitions of the fractional order and the distributed order derivatives [

For any

The Laplace transforms a fractional derivative of Caputo

The distributed derivative of a continuous function

The Laplace transform of the distributed derivative is given by:

Let

The dynamics of the generalized van der Pol oscillator distributed order with complex parameters are studying in this section. We test the intervals of the parameters where there are chaotic and hyperchaotic approaches to the system.

The real form of system (

System (

Therefore, if

To show the solution’s behavior and to obtain the intervals for chaotic and hyperchaotic phenomena of system (

System (

Signs from exponents of Lyapunov and the corresponding solution form.

Dynamics | ||||
---|---|---|---|---|

− | − | − | − | Solution goes to equilibrium point |

0 | − | − | − | Periodic solution (limit cycles) |

0 | 0 | − | − | Quasiperiodic solution (2-torus) |

0 | 0 | 0 | − | Quasiperiodic solution (3-torus) |

+ | − | − | − | Chaotic behavior |

+ | + | − | − | Hyperchaotic behavior of order 2 |

+ | + | + | − | Hyperchaotic behavior of order 3 |

Assuming that

Case 1: Fix

Lyapunov exponents of system (

Lyapunov exponents of system (

Lyapunov exponents of system (

Lyapunov exponent of system (

Case 1: Fix | |||||

Dynamics | |||||

26 | 6.0041 | 3.1382 | −6.1039 | −6.7879 | Hyperchaotic behavior of order 2 |

28 | 9.3759 | 3.7670 | −4.5096 | −6.2100 | Hyperchaotic behavior of order 2 |

30 | 7.4983 | 4.1911 | −5.3663 | −7.3997 | Hyperchaotic behavior of order 2 |

Case 2: Fix | |||||

Dynamics | |||||

21 | 8.0825 | 4.8029 | −10.4143 | −11.0868 | Hyperchaotic behavior of order 2 |

25 | 7.4983 | 4.1911 | −5.3663 | −7.3997 | Hyperchaotic behavior of order 2 |

27 | 8.7146 | 4.6502 | −4.9561 | −6.7303 | Hyperchaotic behavior of order 2 |

Case 3: Fix | |||||

Dynamics | |||||

0.7 | 1.9130 | −0.0303 | −10.6506 | −11.3279 | Chaotic behavior |

2 | 7.4983 | 4.1911 | −5.3663 | −7.3997 | Hyperchaotic behavior of order 2 |

9 | 7.1457 | 3.4489 | −6.1396 | −7.6788 | Hyperchaotic behavior of order 2 |

From Table

Case 2: Fix

This implies our system (

Case 3: Fix

We applied the linear feedback control method to transform the system’s chaotic and hyperchaotic solution (

The distributed order hyperchaotic complex generalized van der Pol system with control (11): (a)

The hyperchaotic complex distributed by order generalized van der Pol system with control (11): (a)

In this section, using active control and Laplace transform, we present a theorem for achieving antisynchronization among the two same systems of (

The drive system (

The system of error can be written as

The antisynchronization between the master system (

Using the control functions (

By transforming system (

By using Theorem

Numerically, if we take

The hyperchaotic solution of system (

Synchronization of the state variables of the master system (

Synchronization error of the master system (

In this work, we have investigated a new generalized van der Pol oscillator distributed order with a complex parameter (

The hyperchaotic solution of system (

The hyperchaotic solution of system (

The authors declare that all data sources are original.

The authors declare no conflicts of interest.

The authors are thankful to the deanship of scientific research Princess Nourah Bint Abdulrahman University, for supporting and funding the work through the research funding program under grant number (FRP144028).