Fig 1.
Chaotic oscillator circuit diagram.
Table 1.
Using the Wolf approach, LEs of system (1) for observation periods (T) = 100, sampling time Δt = 0.5, a = −1.25, b = −0.68, c = 5, , and ω = 0.6 are calculated.
Table 2.
Using the Wolf approach, LEs of system (1) for observation periods (T) = 100, sampling time Δt = 0.5, a = −1.25, b = −0.68, F = 3.28, , ω = 0.6 are calculated.
Fig 2.
Lyapunov exponent for the circuit (1) with F = 3.25.
(a) Lyapunov exponent for circuit (1) with F = 3.25 and initial condition (x, y) = (0.1, 0.1) (b) Lyapunov exponent for circuit (1) with F = 3.25 and initial condition (x, y) = (0.1, 0.5) (c) Lyapunov exponent for circuit (1) with F = 3.25 and initial condition (x, y) = (0.5, 0.1).
Fig 3.
The Lyapunov exponent for the circuit (1) with F = 3.50.
(a) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.1, 0.1) (b) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.5, 0.1) (c) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.5, 0.5) (d) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.01, 0.01) (e) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.05, 0.01) (f) Lyapunov exponent for circuit (1) with F = 3.50 and initial condition (x, y) = (0.01, 0.05).
Fig 4.
Bifurcation diagram for parameter F.
(a) Bifurcation of parameter F with initial condition (x, y) = (0.1, 0.1) (b) Bifurcation of parameter F with initial condition (x, y) = (0.5, 0.1) (c) Bifurcation of parameter F with initial condition (x, y) = (0.1, 0.5) (d) Bifurcation of parameter F with initial condition (x, y) = (0.5, 0.5).
Fig 5.
Bifurcation diagram for the parameters c and μ.
(a) Bifurcation of parameter c with initial condition (x, y) = (0.5, 0.5) (b) Bifurcation of parameter c with initial condition (x, y) = (0.5, 0.5) (c) Bifurcation for parameter c with the initial condition (x, y) = (0.1, 0.1) (d) Bifurcation for parameter μ with initial condition (x, y) = (0.1, 0.1) (e) Bifurcation for parameter F with the initial condition (x, y) = (0.1, 0.5) and the largest Lyapunov exponent (f) Standard Lyapunov stability plot for system (1) where two control parameters F and b vary simultaneously. Red symbolizes low chaos and yellow symbolizes higher chaos.
Fig 6.
Poincaré map for the parameter F.
(a) Poincaré map for parameter F = 1.28 with initial condition (x, y) = (0.1, 0.1) (b) Poincaré map for parameter F = 2.0 with initial condition (x, y) = (0.1, 0.1) (c) Poincaré map for parameter F = 3.25 with initial condition (x, y) = (0.1, 0.1) (d) Poincaré map for parameter F = 3.5 with initial condition (x, y) = (0.1, 0.1) (e) Poincaré map for parameter F = 3.75 with initial condition (x, y) = (0.1, 0.1) (f) Power Spectral Density F = 3.75 with the initial condition (x, y) = (0.1, 0.1).
Fig 7.
Phase portrait for the chaotic attractors.
(a) Phase Portrait for parameter F = 0 with initial condition (x, y) = (0.1, 0.1) (b) Phase Portrait for the parameter F = 1.75 with initial condition (x, y) = (0.1, 0.1) (c) Phase Portrait for the parameter F = 1.78 with initial condition (x, y) = (0.1, 0.1) (d) Phase Portrait for the parameter F = 2.28 with initial condition (x, y) = (0.1, 0.5) (e) Phase Portrait for the parameter F = 2.28 with initial condition (x, y) = (0.5, 0.1) (f) Phase Portrait for the parameter F = 2.28 with initial condition (x, y) = (0.5, 0.5).
Fig 8.
Phase portrait for the chaotic attractors.
(a) Phase Portrait for the parameters F = 0.2 with initial condition (x, y) = (0.1, 0.1) (b) Phase Portrait for parameter F = 0.2 with initial condition (x, y) = (0.1, 0.1) (c) Phase Portrait for parameter F = 0.25 with initial condition (x, y) = (0.1, 0.1) (d) Phase Portrait for parameter F = 0.25 with initial condition (x, y) = (0.1, 0.5).
Fig 9.
Phase portrait with local Lyapunov exponent.
(a) Phase Portrait for the parameter a = 1.4, b = 0.3, c = 3.1, μ = 1.0, ω = 1.0, w = 1.0, and F = 0.2 with initial condition (x, y) = (0.1, 0.1) (b) Phase Portrait for the parameter a = 1.4, b = 0.3, c = 3.1, μ = 1.0, ω = 1.0, w = 1.0, and F = 0.2 with initial condition (x, y) = (0.1, 0.1) (c) Phase Portrait for the parameter a = −1.25, b = −0.68, c = 3.1, , ω = 1.0, F = 0.2, and w = 1.0 with initial condition (x, y) = (0.1, 0.1) (d) Phase Portrait for the parameter a = −1.25, b = −0.68, c = 3.1,
, ω = 1.0, F = 0.2, and w = 1.0 with initial condition (x, y) = (0.1, 0.1).
Fig 10.
Portrait for the periodic, chaotic and resonance nature of the proposed system.
(a) Phase Portrait for the parameter F = 0.2 with initial condition (x, y) = (0.1, 0.1) (b) Phase Portrait for the parameter F = 0.2 with initial condition (x, y) = (0.1, 0.1) (c) Phase Portrait for the parameter F = 0.25 with initial condition (x, y) = (0.1, 0.1) (d) Phase Portrait for the parameter F = 0.25 with initial condition (x, y) = (0.1, 0.5).
Fig 11.
Stabilisation under backstepping control.
Fig 12.
Experimental execution results in plain text.
Fig 13.
Scatter Plot analysis for text data.
(a) Scatter Plot for Text—1 (b) Scatter Plot for Text—2 (c) Scatter Plot for Text—3.
Table 3.
Correlation coefficient between plain text and encrypted text.
Fig 14.
Histogram analysis for text data.
(a) Histogram for Text—1 (b) Histogram for Text—2 (c) Histogram for Text—3.
Table 4.
Histogram variance for plain and cipher text.
Table 5.
Information entropy for plain and cipher text.
Table 6.
Key sensitivity for text encryption.
Fig 15.
Schematic encryption processes.
Fig 16.
Encryption decryption algorithm experimental data set images.
(a) Aeroplane (b) Aeroplane—Encryption (c) Baboon (d) Baboon—Encryption (e) Boat (f) Boat—Encryption (g) House (h) House—Encryption (i) Pepper (j) Pepper—Encryption.
Fig 17.
Correlation distribution for plain and cipher images.
(a) Airplane (b) Pixel gray value on (x, y)—Plain Image (c) Pixel gray value on (x, y)—Cipher Image (d) Baboon (e) Pixel gray value on (x, y)—Plain Image (f) Pixel gray value on (x, y)—Cipher Image (g) Boat (h) Pixel gray value on (x, y)—Plain Image (i) Pixel gray value on (x, y)—Cipher Image (j) House (k) Pixel gray value on (x, y)—Plain Image (l) Pixel gray value on (x, y)—Cipher Image (m) Pepper (n) Pixel gray value on (x, y)—Plain Image (o) Pixel gray value on (x, y)—Cipher Image.
Table 7.
Plain and cipher image correlation coefficients.
Fig 18.
Histogram analysis for plain and encrypted images.
(a) Histogram Airplane—Plain Image (b) Histogram Airplane—Cipher Image (c) Histogram Baboon—Plain Image (d) Histogram Baboon—Cipher Image (e) Histogram Boat—Plain Image (f) Histogram Boat—Cipher Image (g) Histogram House—Plain Image (h) Histogram House—Cipher Image (i) Histogram Pepper—Plain Image (j) Histogram Pepper—Cipher Image.
Table 8.
Histogram variance for plain and cipher images.
Table 9.
Entropy value for plain and cipher images.
Table 10.
NPCR and UACI values.
Fig 19.
Decryption effects of cipher images when making small changes in the key parameters.
(a) Airplane Key with (x0 = 0.1, y = 0.5, F = 2) (b) Airplane Key with (x0 = 0.8, y = 0.5, F = 2.5) (c) Baboon Key with (x0 = 0.8, y = 0.5, b = 1.3) (d) Baboon Key with (x0 = 0.4, y = 0.1, F = 3.5) (e) Boat Key with (x0 = 0.3, y = 0.8, b = 2.5) (f) Boat Key with (x0 = 0.1, y = 0.5, F = 8) (g) House Key with (x0 = 0.1, y = 0.5, F = 0.5) (h) House Key with (x0 = 0.1, y = 0.1, b = 8.5) (i) Pepper Key with (x0 = 1.5, y = 2.5, b = 10.5) (j) Pepper Key with (x0 = 3.5, y = 2.5, F = 12).
Table 11.
Approximate encrypted pixels are different between the two encrypted Images.
Fig 20.
Results of data loss attacks, retrieved effectively from the defective cipher image.
(a) Airplane Losing data by 20% (b) Baboon Losing data by 30% (c) Boat Losing data by 40% (d) Pepper Losing data by 50% (e) Airplane Retrieval data by 20% (f) Baboon Retrieval data by 30% (g) Boat Retrieval data by 40% (h) Pepper Retrieval data by 50%.
Table 12.
Lossing plain and cipher image MSE and PSNR details.
Table 13.
MSE and PSNR value for plain and decrypted images.
Table 14.
Comparison of different image cryptosystems.