Fig 1.
PWL function in a chaotic oscillator based on: (a) SNLF series, (b) negatives slopes, and (c) sawtooth function.
Table 1.
19 chaotic oscillators proposed by J. C. Sprott [20].
Table 2.
Equilibrium points and eigenvalues of the three chaotic oscillators based on PWL functions.
Table 3.
Equilibrium points and eigenvalues of Sprott’s collection.
Table 4.
Lyapunov exponent and fractal dimension of each state variable x1, x2, x3 of the twenty two chaotic oscillators computed by “TISEAN 3.0.1”.
Table 5.
Lyapunov exponents and fractal dimension of the twenty two chaotic oscillators avaluated by “Wolf’s algorithm”.
Fig 2.
Block diagram of the master-slave chaotic synchronization applying Pecora-Carroll technique by using the state variable x1 as the driving signal.
Fig 3.
Phase diagrams for the master and slave state variables: (a) x2 and (b) x3 for Sprott’s case L.
Fig 4.
Synchronization error using x1 as drive for Sprott’s case L, for the master and slave state variables x2 and x3.
Fig 5.
Phase diagrams for the master and slave state variables: (a) x1, (b) x2 and (c) x3 for Sprott’s case L applying Hamiltonian forms and observer approach.
Fig 6.
Synchronization error of Sprott’s case L, for the master and slave state variables applying Hamiltonian forms.
Fig 7.
Phase diagrams for the master and slave state variables: (a) x1, (b) x2 and (c) x3 for Sprott’s case L applying OPCL technique with P = −3 in (25).
Fig 8.
Synchronization error of Sprott’s case L, for the master and slave state variables applying OPCL technique.
Table 6.
FPGA resources of the chaotic oscillator based on SNLF series, and Sprott’s cases G and L by applying three numerical methods and using Cyclone IV GX EP4CGX150DF31C7.
Fig 9.
Experimental phase-space portraits x1 − x2 of the chaotic oscillators: (a) Based on SNLF series with axes X = 2V/div and Y = 1V/div, (b) Sprott’s case G with axes X = 1V/div and Y = 1V/div, and (c) Sprott’s case L with axes X = 1V/div and Y = 1V/div.
Fig 10.
Experimental synchronization of x2 (left column) and x3 (right column) phase diagrams applying Pecora-Carroll synchronization technique to the chaotic oscillator: (a) based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
In all cases with axes channels: X = 1V/div and Y = 1V/div.
Fig 11.
Experimental synchronization of x1 (left column), x2 (center column), and x3 (right column) phase diagrams applying Hamiltonian forms to the chaotic oscillator: (a) based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
In all cases with axes channels: X = 1V/div and Y = 1V/div.
Fig 12.
Experimental synchronization of x1 (left column), x2 (center column), and x3 (right column) phase diagrams applying OPCL to the chaotic oscillator: (a) based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
In all cases with axes channels: X = 1V/div and Y = 1V/div.
Table 7.
Resources for the three synchronization techniques applied to the chaotic oscillator based on SNLF series, and Sprott’s cases G and L, using the FPGA Cyclone IV GX EP4CGX150DF31C7.
Fig 13.
Experimental phase-space portraits x1 − x2 of the chaotic oscillators: (a) Based on Negative Slopes, (b) Sprott’s case B, and (c) Sprott’s case S.
In all cases with axes X = 1V = div and Y = 1V = div.
Fig 14.
Experimental phase diagrams applying Pecora-Carroll synchronization technique to the chaotic oscillator: (a) based on Negative Slopes (x2 = left column and x3 = right column), (b) Sprott’s case B (x1 = left column and x3 = right column), and (c) Sprott’s case S (x1 = left column and x2 = right column).
In all cases with axes channels: X = 2V = div and Y = 2V = div.
Fig 15.
Experimental phase diagrams of the master-slave state variables x1 (left column), x2 (center column), and x3 (right column), applying Hamiltonian forms to the chaotic oscillator: (a) based on Negative Slopes, (b) Sprott’s case B, and (c) Sprott’s case S.
In all cases with axes channels: X = 2V = div and Y = 2V = div.
Fig 16.
Experimental phase diagrams of the master-slave state variables x1 (left column), x2 (center column), and x3 (right column), applying OPCL to the chaotic oscillator: (a) based on Negative Slopes, (b) Sprott’s case B, and (c) Sprott’s case S.
In all cases with axes channels: X = 2V = div and Y = 2V = div.
Fig 17.
Chaotic secure communication system based on the master-slave topology.
Fig 18.
Block diagram of the implementation of a chaotic secure communication system based on a master-slave topology.
Fig 19.
Original (left column), encrypted (center column), and recovered (right column) images applying Pecora-Carroll synchronization technique to the chaotic oscillator: (a) Based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
Fig 20.
Original (left column), encrypted (center column), and recovered (right column) images applying Hamiltonian forms to the chaotic oscillator: (a) Based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
Fig 21.
Original (left column), encrypted (center column), and recovered (right column) images applying OPCL technique to the chaotic oscillator: (a) Based on SNLF series, (b) Sprott’s case G, and (c) Sprott’s case L.
Table 8.
Correlations among the original, encrypted and recovered data using the three synchronization techniques and the oscillators with the highest positive Lyapunov exponent obtained with “TISEAN 3.0.1”.
Fig 22.
Original (left column), encrypted (center column), and recovered (right column) images applying Pecora-Carroll synchronization technique to the chaotic oscillator: (a) Based on Negative Slopes (Variable x2), (b) Sprott’s case B (Variable x1), and (c) Sprott’s case S (Variable x1).
Fig 23.
Original (left column), encrypted (center column), and recovered (right column) images applying Hamiltonian forms to the chaotic oscillator: (a) Based on Negative Slopes (Variable x1), (b) Sprott’s case B (Variable x1), and (c) Sprott’s case S (Variable x2).
Fig 24.
Original (left column), encrypted (center column), and recovered (right column) images applying OPCL to the chaotic oscillator: (a) Based on Negative Slopes (Variable x3), (b) Sprott’s case B (Variable x1), and (c) Sprott’s case S (Variable x3).
Table 9.
Correlations among the original, encrypted and recovered data using the three synchronization techniques and the oscillators with the high positive Lyapunov exponent values obtained with “Wolf’s algorithm”.