A Logistic Map Runge Kutta-4 Solution for FPGA Using Fixed Point Representation

Year 2019, Volume: 1 Issue: 1, 19 - 28, 30.11.2019

Emre Güngör Enver Cavus İhsan Pehlivan

Abstract

Logistic map can show simple chaotic behavior pattern. Chaotic patterns with their irregularity and unpredictability properties are often used for random number generation and encryption. Hence, simple implementation of logistic map with chaotic behaviour is studied on how to solve differential equations with Runge Kutta Order-4 on FPGA and compared with floating point and fixed point representations. In the study, the logistic map equation is modelled with Verilog hardware description language and system is simulated using ModelSim and MATLAB environment. This paper reveals fundemental chaotic pattern implementation using logistic map to obtain unpredictable resuts on FPGA using xed point number representation. Experimental results confirm both Verilog and MATLAB implementation produce same results. The results obtained from system in different generations can be converted to fixed point numeric values with an error margin. Error margin changes based on bit-length of fixed point number representation design.

Keywords

Chaos, Logistic Map, FPGA, Verilog

References

• [1] N.K. Pareek, Vinod Patidar, K.K. Sud, Image encryption using chaotic logistic map Image and Vision Computing Vol. 24, Issue 9, pp. 926-934, 1 Sept. 2006.
• [2] Kocarev, Ljupo, and Goce Jakimoski. "Logistic map as a block encryption algorithm." Physics Letters A 289.4, pp. 199-206, 2001.
• [3] K.H. Tsoi, K.H. Leung , and P.H.W. Leong, Compact FPGA-based True and Pseudo Random Number Generators Field-Programmable Custom Computing Machines, FCCM 2003, 11th Annual IEEE Symposium, pp. 51-61 ,9-11 April 2003.
• [4] Phatak, S. C., and S. Suresh Rao. "Logistic map: A possible random-number generator." Physical review E 51.4, 3670, 1995.
• [5] Aseeri M.A, Shobhy M.I., Lee P., Lorenz chaotic model using Filed Programmable Gate Array(FPGA), 45th Midwest Symposium on Circuits and Systems, MWSCAS-2002, vol. 1, pp. I-527-30, 4-7 Aug. 2002.
• [6] Azzaz M.S, Tanougast C., Sadoudi S., Dandache A., Real-time FPGA implementation of Lorenzs chaotic generator for ciphering telecommunications, Circuits and Systems and TAISA Conference, NEWCAS-TAUSA 09, pp. 1-4, 2009.
• [7] May, Robert M. "Simple mathematical models with very complicated dynamics." Nature 261.5560,pp 459-467, 1976.
• [8] Modern Automation Systems, Laxmi Publications, Ltd., 01-May-2009.
• [9] Granberg, T. , Handbook of digital techniques for high-speed design: design examples, signaling and memory technologies, fiber optics, modeling and simulation to ensure signal integrity., Prentice-Hal, 2004.
• [10] Hall C.D., Manifesto on Numerical Integration of Equations of Motion Using Matlab, 2002.
• [11] MATLAB, The MathWorks, Inc., Natick, Massachusetts, United States.
• [12] Eibl, J. "http://kdiff3.sourceforge.net/". 2003.