An interval chaos insight to iterative decomposition method for Rossler differential equation by considering stable uncertain coefficients

Document Type : Research Paper


1 Department of Electrical Engineering, Tafresh University, Tafresh 39518-79611, Iran.

2 Departments of Mathematics, Tafresh University, Tafresh 39518-79611, Iran.


Generally, in most applications of engineering, the parameters of the mathematical models are considered deterministic. Although, in practice, there are always some uncertainties in the model parameters; these uncertainties may be made wrong representation of the mathematical model of the system. These uncertainties can be generated from different reasons like measurement error, inhomogeneity of the process, chaotic behavior of systems, etc. This problem leads researchers to study these uncertainties and propose solutions for this problem. The iterative analysis is a method that can be utilized to solve these kinds of problems. In this paper, a new combined method based on interval chaotic and iterative decomposition method is proposed. The validation of the proposed method is performed on a chaotic Rossler system in stable Intervals. The simulation results are applied on 2 practical case studies and the results are compared with the interval Chebyshev method and RungeKutta method of order four (RK4) method. The final results showed that the proposed method has a good performance in finding the confidence interval for the Rossler models with interval uncertainties; the results also showed that the proposed method can handle the wrapping effect in a better manner to sharpen the range of non-monotonic interval.


  • [1]          O. Abdulaziz, N. F. M. Noor, I. Hashim, and M. S. M. Noorani, Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos, Solitons and Fractals, 36 (2008), 14051411.
  • [2]          H. N. Agiza and M. T. Yassen, Synchronization of Rossler and Chen chaotic dynamical systems using active control, Physics Letters A, 4 (2001), 191–197.
  • [3]          E. Celik, M. Bayram, and T. Yelolu, Solution of Differential-Algebraic Equations (DAEs) by Adomian Decompo- sition Method, International Journal Pure and Applied Mathematical Sciences, 3(1) (2006), 93–100.
  • [4]          Sh. Chen and Su. Huan, and J. Wu, Interval optimization of dynamic response for structures with interval parameters, Computers and structures, 82(1) (2004), 1–11.
  • [5]          DJ. Evans and K. Raslan, The Adomian decomposition method for solving delay differential equation, International Journal and Computer Mathematics, 82(1) (2004), 914–923.
  • [6]          G. Gaxiola, J. A. Santiago, and J. Ruiz de Chvez, Solution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method, (2016).
  • [7]          S. M. Goh, M. S. M. Noorani, and I. Hashim, A new application of variational iteration method for the chaotic Rossler system, Chaos, Solitons and Fractals, 42 (2009), 604–1610.
  • [8]          I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail, and A. M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, Solitons and Fractals, 28 (2006), 1149–1158.
  • [9]          M. M. Hosseini, Adomian decomposition method with Chebyshev polynomials, Applied Mathematical and Com- putation, 175 (2016), 1685–1693.
  • [10]        H. Jafari and V. Daftardar Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, Journal of Computational and Applied Mathematics, 196 (2006), 644–651.
  • [11]        Zhu. Jiandong and Yu. Ping Tian, Stabilizing periodic solutions of nonlinear systems and applications in chaos control, Canad. J. Math, 16 (1964), 539–548.
  • [12]        B. Lazhar, A. Majid Wazwaz, and R. Rach, Dual solutions for nonlinear boundary value problems by the Adomian decomposition method, International Journal of Numerical Methods for Heat and Fluid Flow, 26(8) (2016).
  • [13]        D. Lesnic, Convergence of Adomian decomposition method: periodic temperatures, Comput. Math. Appl, 44 (2002), 13–24.
  • [14]        M. Mossa Al-Sawalha, M. S. M. Noorani, and I. Hashim, On accuracy of Adomian decomposition method for hyperchaotic Rossler system, Chaos, Solitons & Fractals, 40(4) (2009), 1801–1807.
  • [15]        K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics letters A, 6 (1992), 421–428.
  • [16]        N. Razmjooy and M. Ramezani, Analytical Solution for Optimal Control by the Second kind Chebyshev Polyno- mials Expansion, Iranian Journal of Science and Technology, (2016).
  • [17]        S. Tangaramvong, F. Tin-Loi, C. Yang, and W. Gao, Interval analysis of nonlinear frames with uncertain con- nection properties International Journal of Non-Linear Mechanics, (2016), 83–95.
  • [18]        M. Tatari, M. Dehghan, and M. Razzaghi, Application of the Adomian Decomposition Method for the Fokker-  Planck Equation, Mathematical and Computer Modelling, 45(5) (2007), 639–650.
  • [19]        D. B. West and D. B. Shmoys, Recognizing graph with fixed interval number is NP-complete, Discrete Applied Mathematics, 8 (1981), 295–305.
  • [20]        J. Wu, J. Gao, Z. Luo, and B. T. Robust, Topology optimization for structures under interval uncertainty, Advances in Engineering Software, (2016), 36–48.