The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications.
Citation: Xinyu Pan. Research on discrete differential solution methods for derivatives of chaotic systems[J]. AIMS Mathematics, 2024, 9(12): 33995-34012. doi: 10.3934/math.20241621
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The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications.
The definition of impulsive semi-dynamical system and its properties including the limit sets of orbits have been investigated [1,9]. The generalized planar impulsive dynamical semi-dynamical system can be described as follows
{dxdt=P(x,y),dydt=Q(x,y),(x,y)∉M,△x=a(x,y),△y=b(x,y),(x,y)∈M, | (1) |
where
I(z)=z+=(x+,y+)∈R2, x+=x+a(x,y), y+=y+b(x,y) |
and
Let
C+(z)={Π(z,t)|t∈R+} |
is called the positive orbit of
M+(z)=C+(z)∩M−{z}. |
Based on above notations, the definition of impulsive semi-dynamical system is defined as follows [1,9,23].
Definition 1.1. An planar impulsive semi-dynamic system
F(z,(0,ϵz))∩M=∅ and Π(z,(0,ϵz))∩M=∅. |
Definition 1.2. Let
1.
2. for each
It is clear that
Definition 1.3. Let
Denote the points of discontinuity of
Theorem 1.4. Let
In 2004 [2], the author pointed out some errors on Theorem 1.4, that is, it need not be continuous under the assumptions. And the main aspect concerned in the paper [2] is the continuality of
In the following we will provide an example to show this Theorem is not true for some special cases. Considering the following model with state-dependent feedback control
{dx(t)dt=ax(t)[1−x(t)K]−βx(t)y(t)1+ωx(t),dy(t)dt=ηβx(t)y(t)1+ωx(t)−δy(t),}x<ET,x(t+)=(1−θ)x(t),y(t+)=y(t)+τ,}x=ET. | (2) |
where
Define four curves as follows
L0:x=δηβ−δω; L1:y=rβ[1−xK](1+ωx); |
L2:x=ET; and L3:x=(1−θ)ET. |
The intersection points of two lines
yET=rβ[1−ETK](1+ωET), yθET=rβ[1−(1−θ)ETK](1+ω(1−θ)ET). |
Define the open set in
Ω={(x,y)|x>0,y>0,x<ET}⊂R2+={(x,y)|x≥0,y≥0}. | (3) |
In the following we assume that model (2) without impulsive effects exists an unstable focus
E∗=(xe,ye)=(δηβ−δω,rη(Kηβ−Kδω−δ)K(ηβ−δω)2), |
which means that model (2) without impulsive effects has a unique stable limit cycle (denoted by
In the following we show that model (2) defines an impulsive semi-dynamical system. From a biological point of view, we focus on the space
Further, we define the section
y+k+1=P(y+k)+τ=y(t1,t0,(1−θ)ET,y+k)+τ≐PM(y+k), and Φ(y+k)=t1. | (4) |
Now define the impulsive set
M={(x,y)| x=ET,0≤y≤YM}, | (5) |
which is a closed subset of
N=I(M)={(x+,y+)∈Ω| x+=(1−θ)ET,τ≤y+≤P(yθET)+τ}. | (6) |
Therefore,
According to the Definition 1.3 and topological structure of orbits of model (2) without impulsive effects, it is easy to see that
However, this is not true for case (C) shown in Fig. 2(C). In fact, for case (C) there exists a trajectory (denoted by
If we fixed all the parameter values as those shown in Fig. 3, then we can see that the continuities of the Poincaré map and the function
Theorem 2.1. Let
Note that the transversality condition in Theorem 2.1 may exclude the case (B) in Fig. 2(B). In fact, based on our example we can conclude that the function
Recently, impulsive semi-dynamical systems or state dependent feedback control systems arise from many important applications in life sciences including biological resource management programmes and chemostat cultures [5,6,10,12,17,18,19,20,21,22,24], diabetes mellitus and tumor control [8,13], vaccination strategies and epidemiological control [14,15], and neuroscience [3,4,7]. In those fields, the threshold policies such as
The above state-dependent feedback control strategies can be defined in broad terms in real biological problems, which are usually modeled by the impulsive semi-dynamical systems. The continuity of the function
[1] |
S. Zhou, X. Wang, Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations, Chaos Soliton. Fract., 139 (2020), 109981. https://doi.org/10.1016/j.chaos.2020.109981 doi: 10.1016/j.chaos.2020.109981
![]() |
[2] |
Z. Z. Ma, Q. C. Yang, R. P. Zhou, Lyapunov exponent algorithm based on perturbation theory for discontinuous systems, Acta Phys. Sin., 70 (2021), 240501. https://doi.org/10.7498/aps.70.20210492 doi: 10.7498/aps.70.20210492
![]() |
[3] |
F. Nazarimehr, S. Panahi, M. Jalili, M. Perc, S. Jafari, B. Fercec, Multivariable coupling and synchronization in complex networks, Appl. Math. Comput., 372 (2020), 124996. https://doi.org/10.1016/j.amc.2019.124996 doi: 10.1016/j.amc.2019.124996
![]() |
[4] |
N. Zandi-Mehran, S. Jafari, S. M. R. H. Golpayegani, Signal separation in an aggregation of chaotic signals, Chaos Soliton. Fract., 138 (2020), 109851. https://doi.org/10.1016/j.chaos.2020.109851 doi: 10.1016/j.chaos.2020.109851
![]() |
[5] |
S. J. Cang, L. Wang, Y. P. Zhang, Z. Wang, Z. Chen, Bifurcation and chaos in a smooth 3D dynamical system extended from Nosé-Hoover oscillator, Chaos Soliton. Fract., 158 (2022), 112016. https://doi.org/10.1016/j.chaos.2022.112016 doi: 10.1016/j.chaos.2022.112016
![]() |
[6] |
V. V. Klinshov, V. A. Kovalchuk, I. Franović, M. Perc, M. Svetec, Rate chaos and memory lifetime in spiking neural networks, Chaos Soliton. Fract., 158 (2022), 112011. https://doi.org/10.1016/j.chaos.2022.112011 doi: 10.1016/j.chaos.2022.112011
![]() |
[7] |
K. D. S. Andrade, M. R. Jeffrey, R. M. Martins, M. A. Teixeira, Homoclinic boundary-saddle bifurcations in planar nonsmooth vector fields, Int. J. Bifurcat. Chaos, 32 (2022), 22300099. https://doi.org/10.1142/S0218127422300099 doi: 10.1142/S0218127422300099
![]() |
[8] |
N. Yadav, S. Shah, Topological weak specification and distributional chaos on noncompact spaces. Int. J. Bifurcat. Chaos, 32 (2022), 2250048. https://doi.org/10.1142/S0218127422500481 doi: 10.1142/S0218127422500481
![]() |
[9] |
X. Y. Pan, H. M. Zhao, Research on the entropy of logistic chaos, Acta Phys. Sin., 61 (2012), 200504. https://doi.org/10.7498/aps.61.200504 doi: 10.7498/aps.61.200504
![]() |
[10] |
H. P. Wen, S. M. Yu, J. H. Lü, Encryption algorithm based on Hadoop and non-degenerate high-dimensional discrete hyperchaotic system, Acta Phys. Sin., 66 (2017), 230503. https://doi.org/10.7498/aps.66.230503 doi: 10.7498/aps.66.230503
![]() |
[11] |
X. Y. Wan, J. M. Zhang, A novel image authentication and recovery algorithm based on dither and chaos, Acta Phys. Sin., 63 (2014), 210701. https://doi.org/10.7498/aps.63.210701 doi: 10.7498/aps.63.210701
![]() |
[12] |
B. Yang, X. Liao, Some properties of the Logistic map over the finite field and its application, Signal process., 153 (2018), 231–242. https://doi.org/10.1016/j.sigpro.2018.07.011 doi: 10.1016/j.sigpro.2018.07.011
![]() |
[13] |
M. Lazaros, V. Christos, J. Sajad, J. M. Munoz-Pacheco, J. Kengne, K. Rajagopal, et al., Modification of the logistic map using fuzzy numbers with application to pseudorandom number generation and image encryption, Entropy, 22 (2020), 474. https://doi.org/10.3390/e22040474 doi: 10.3390/e22040474
![]() |
[14] |
M. Wang, X. Wang, T. Zhao, C. Zhang, Z. Xia, N. Yao, Spatiotemporal chaos in improved cross coupled map lattice and its application in a bit-level image encryption scheme, Inform. Sciences, 554 (2021), 1–24. https://doi.org/10.1016/j.ins.2020.07.051 doi: 10.1016/j.ins.2020.07.051
![]() |
[15] |
X. Y. Wang, S. Gao, X. L. Ye, S. Zhou, M. X. Wang, A new image encryption algorithm with cantor diagonal scrambling based on the PUMCML system, Int. J. Bifurcat. Chaos, 31 (2021), 2150003. https://doi.org/10.1142/S0218127421500036 doi: 10.1142/S0218127421500036
![]() |
[16] |
Z. P. Zhao, S. Zhou, X. Y. Wang, A new chaotic signal based on deep learning and its application in image encryption, Acta Phys. Sin., 70 (2021), 230502. https://doi.org/10.7498/aps.70.20210561 doi: 10.7498/aps.70.20210561
![]() |
[17] |
B. X. Mao, Two methods contrast of sliding mode synchronization of fractional-order multy-chaotic systems, Acta Electronica Sin., 48 (2020), 2215–2219. https://doi.org/10.3969/j.issn.0372-2112.2020.11.017 doi: 10.3969/j.issn.0372-2112.2020.11.017
![]() |
[18] |
B. X. Mao, D. X. Wang. Self-adaptive sliding mode synchronization of uncertain fractional-order high-dimension chaotic systems, Acta Electronica Sin., 49 (2021), 775–780. https://doi.org/10.12263/DZXB.20200316 doi: 10.12263/DZXB.20200316
![]() |
[19] |
Z. C. Zhu, Q. X. Zhu, Adaptive neural prescribed performance control for non-triangular structural stochastic highly nonlinear systems under hybrid attacks, IEEE T. Automat. Sci. Eng., 2024. https://doi.org/10.1109/TASE.2024.3447045 doi: 10.1109/TASE.2024.3447045
![]() |
[20] |
Q. X. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lexvy processes, IEEE T. Automat. Control, 2024. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
![]() |
[21] |
Y. Xue, J. Han, Z. Tu, X. Y. Chen, Stability analysis and design of cooperative control for linear delta operator system, AIMS Math., 8 (2023), 12671–12693. https://doi.org/10.3934/math.2023637 doi: 10.3934/math.2023637
![]() |
[22] |
H. Bi, G. Qi, J. Hu, P. Faradja, G. Chen, Hidden and transient chaotic attractors in the attitude system of quadrotor unmanned aerial vehicle, Chaos Soliton. Fract., 138 (2020), 109815. https://doi.org/10.1016/j.chaos.2020.109815 doi: 10.1016/j.chaos.2020.109815
![]() |
[23] |
L. X. Fu, S. B. He, H. H. Wang, K. H. Sun, Simulink modeling and dynamic characteristics of discrete memristor chaotic system, Acta Phys. Sin., 71 (2022), 030501. https://doi.org/10.7498/aps.71.20211549 doi: 10.7498/aps.71.20211549
![]() |
[24] |
J. Y. Ruan, K. H. Sun, J. Mou. Memristor-based Lorenz hyper-chaotic system and its circuit implementation, Acta Phys. Sin., 65 (2016), 190502. https://doi.org/10.7498/aps.65.190502 doi: 10.7498/aps.65.190502
![]() |
[25] |
J. V. N. Tegnitsap, H. B. Fotsin, Multistability, transient chaos and hyperchaos, synchronization, and chimera states in wireless magnetically coupled VDPCL oscillators, Chaos Soliton. Fract., 158 (2022), 112056. https://doi.org/10.1016/j.chaos.2022.112056 doi: 10.1016/j.chaos.2022.112056
![]() |
[26] |
H. Xiao, Z. Li, H. Lin, Y. Zhao, A sual rumor spreading model with consideration of fans versus ordinary people, Mathematics, 11 (2023), 2958. https://doi.org/10.3390/math11132958 doi: 10.3390/math11132958
![]() |
[27] |
Q. Yang, X. Wang, X. Cheng, B. Du, Y. Zhao, Positive periodic solution for neutral-type integral differential equation arising in epidemic model, Mathematics, 11 (2023), 2701. https://doi.org/10.3390/math11122701 doi: 10.3390/math11122701
![]() |
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