Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points

Recently, systems with infinite equilibria have attracted interest because they can be considered as systems with hidden attractors. In this work, we introduce a new elegant system with an open curve of equilibrium points. It has just one parameter (a) and an exponential function. We show the dynamics of such a new chaotic oscillator by computing the Lyapunov exponents’ spectrum and bifurcation diagram, and it is implemented by using a field-programmable gate array (FPGA). The exponential function is approached by power series and then implemented with adders and multipliers within the FPGA. Experimental results are provided for the attractor as well as for the synchronization of two chaotic oscillators to transmit an image, thus demonstrating the usefulness of the new oscillator in a chaotic secure communication system.

[1]  M. A. Platas-Garza,et al.  Chaotic synchronization of regular and irregular complex networks with fractional order oscillators , 2016 .

[2]  G. Zhong,et al.  Experimental confirmation of chaos from Chua's circuit , 1985 .

[3]  Taner Tuncer.  The implementation of chaos-based PUF designs in field programmable gate array , 2016 .

[4]  Bocheng Bao,et al.  Inductor-free simplified Chua’s circuit only using two-op-amp-based realization , 2016 .

[5]  Guanrong Chen,et al.  Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications , 2006 .

[6]  Shukai Duan,et al.  A Memristor-Based Chaotic System and Its Field Programmable Gate Array Implementation , 2016 .

[7]  Viet-Thanh Pham,et al.  A Chaotic System with Different Shapes of Equilibria , 2016, Int. J. Bifurc. Chaos.

[8]  Rafael Martínez-Guerra,et al.  A new reduced-order observer for the synchronization of nonlinear chaotic systems: An application to secure communications. , 2015, Chaos.

[9]  Sundarapandian Vaidyanathan,et al.  A no-equilibrium hyperchaotic system with a cubic nonlinear term , 2016 .

[10]  Kehui Sun,et al.  A nonlinear circuit with two memcapacitors , 2016 .

[11]  E. Tlelo-Cuautle,et al.  Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs , 2016 .

[12]  Xiaole Fang,et al.  Study on a new chaotic bitwise dynamical system and its FPGA implementation , 2015 .

[13]  Sundarapandian Vaidyanathan,et al.  Hidden attractors in a chaotic system with an exponential nonlinear term , 2015 .

[14]  Miguel Angel Murillo-Escobar,et al.  A robust embedded biometric authentication system based on fingerprint and chaotic encryption , 2015, Expert Syst. Appl..

[15]  Guanrong Chen,et al.  Constructing hyperchaotic systems at will , 2015, Int. J. Circuit Theory Appl..

[16]  Mauricio Zapateiro,et al.  A modified Chua chaotic oscillator and its application to secure communications , 2014, Appl. Math. Comput..

[17]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[18]  Esteban Tlelo-Cuautle,et al.  VHDL Descriptions for the FPGA Implementation of PWL-Function-Based Multi-Scroll Chaotic Oscillators , 2016, PloS one.

[19]  Sundarapandian Vaidyanathan,et al.  A Chaotic System With Equilibria Located on the Rounded Square Loop and Its Circuit Implementation , 2016, IEEE Transactions on Circuits and Systems II: Express Briefs.

[20]  Enzeng Dong,et al.  Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement , 2016 .

[21]  Vahid Rashtchi,et al.  FPGA Implementation of a Real-Time Weak Signal Detector Using a Duffing Oscillator , 2015, Circuits Syst. Signal Process..

[22]  Viet-Thanh Pham,et al.  Bifurcation analysis and circuit realization for multiple-delayed Wang–Chen system with hidden chaotic attractors , 2016 .

[23]  Esteban Tlelo-Cuautle,et al.  A survey on the integrated design of chaotic oscillators , 2013, Appl. Math. Comput..

[24]  Sundarapandian Vaidyanathan,et al.  A Chaotic System with Different Families of Hidden Attractors , 2016, Int. J. Bifurc. Chaos.

[25]  Ali Kanso,et al.  An efficient and robust image encryption scheme for medical applications , 2015, Commun. Nonlinear Sci. Numer. Simul..

[26]  Luis Gerardo de la Fraga,et al.  Engineering Applications of FPGAs , 2016 .

[27]  Jacques M. Bahi,et al.  Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems , 2015, IEEE Transactions on Circuits and Systems I: Regular Papers.

[28]  Ping Zhou,et al.  Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points , 2014 .

[29]  Jun Ma,et al.  Pattern Selection in Network of Coupled Multi-Scroll Attractors , 2016, PloS one.

[30]  Bharathwaj Muthuswamy,et al.  A Route to Chaos Using FPGAs , 2015 .

[31]  César Cruz-Hernández,et al.  Chaotic synchronization of irregular complex network with hysteretic circuit-like oscillators in hamiltonian form and its application in private communications , 2016 .

[32]  José-Cruz Nuñez Pérez,et al.  FPGA realization of multi-scroll chaotic oscillators , 2015, Commun. Nonlinear Sci. Numer. Simul..

[33]  César Cruz-Hernández,et al.  Coupling strength computation for chaotic synchronization of complex networks with multi-scroll attractors , 2016, Appl. Math. Comput..

[34]  Rafael Martínez-Guerra,et al.  Secure communications via synchronization of Liouvillian chaotic systems , 2016, J. Frankl. Inst..

[35]  Leon O. Chua,et al.  Chua's circuit 10 years later , 1994, Int. J. Circuit Theory Appl..

[36]  Tanmoy Banerjee,et al.  A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation , 2016 .

[37]  Miguel Angel Murillo-Escobar,et al.  Implementation of an improved chaotic encryption algorithm for real-time embedded systems by using a 32-bit microcontroller , 2016, Microprocess. Microsystems.

[38]  Takashi Matsumoto,et al.  A chaotic attractor from Chua's circuit , 1984 .

[39]  Runtong Chu,et al.  Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice , 2014 .