# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021263
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

 1 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China 2 Zhejiang Institute, China University of Geosciences, Hangzhou, Zhejiang 311305, China 3 Beijing University of Technology, Beijing, 100124, China

* Corresponding author: weizhouchao@163.com

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No.11772306, 12172340), Zhejiang Provincial Natural Science Foundation of China under Grant (No.LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05). The last author is supported by National Natural Science Foundation of China (Grant No. 11832002)

In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.

Citation: Zhouchao Wei, Fanrui Wang, Huijuan Li, Wei Zhang. Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021263
##### References:

show all references

##### References:
The phase portraits of hyperchaotic dynamo system with the parameter values for $(m,g,r,k_{1},k_{2}) = (0.04,140.6,7,34,12)$: (a) $x$-$y$ plane; (b) time series of $x(t)$
Phase portraits of periodic orbit: (a) $x$-$y$ plane; (b) $x$-$z$ plane; (c) $x$-$u$ plane; (d) $x$-$v$ plane; (e) time series of $x(t)$
Part of time variation figures for four judgment conditions with $(m,g,r,k_{1},k_{2}) = (0.04,140.6,3.5,34,12)$
Deviation vector near $E_{1}$ under the initial condition $(\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0)$ and different values for $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4})$: (a) $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8})$; (b) $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5})$
Deviation vector near $E_{2}$ under the initial condition $(\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0)$ and different values for $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4})$: (a) $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8})$; (b) $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5})$
Time-variation of instability exponent near the equilibria under the initial condition $(\xi_{1},\xi_{2},\xi_{3},\xi_{4}) = (0,0,0,0)$ and $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-8},10^{-8},10^{-8},10^{-8})$ (left), $(\dot{\xi}_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4}) = (10^{-5},10^{-5},10^{-5},10^{-5})$ (right)
(a) Time series of system (31) with hidden hyperchaotic attractor; (b) The trajectory of the indefinite Lyapunov function $V$ for system (32) with and the resetting time $t_{k} = 0.01 k,\,\, k = 1,2,\cdots$. The initial condition we choose is $(-5.6692,0.1119, -20.3280, -36.3177,-114.4281)$
(a) Time series of system (31) with hidden hyperchaotic attractor; (b) The trajectory of the indefinite Lyapunov function $V$ for system (32) and the resetting time $t_{k} = 0.01 k,\,\, k = 1,2,\cdots.$ The initial condition we choose is $(8,2, 2, 1,1)$
 [1] Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098 [2] Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6053-6069. doi: 10.3934/dcdsb.2019130 [3] Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 945-976. doi: 10.3934/dcdsb.2021076 [4] Gang Tian. Bott-Chern forms and geometric stability. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 211-220. doi: 10.3934/dcds.2000.6.211 [5] C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435 [6] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [7] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [8] Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 [9] Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091 [10] Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527 [11] Yingjing Shi, Rui Li, Honglei Xu. Control augmentation design of UAVs based on deviation modification of aerodynamic focus. Journal of Industrial & Management Optimization, 2015, 11 (1) : 231-240. doi: 10.3934/jimo.2015.11.231 [12] Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 [13] Aram Arutyunov, Dmitry Karamzin, Fernando L. Pereira. On a generalization of the impulsive control concept: Controlling system jumps. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 403-415. doi: 10.3934/dcds.2011.29.403 [14] Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1 [15] Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069 [16] Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317 [17] Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683 [18] Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 [19] Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 [20] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

2020 Impact Factor: 1.327