An active charge-controlled memristive Chua’s circuit is implemented, and its basic properties are analyzed. Firstly, with the system trajectory starting from an equilibrium point, the dynamic behavior of multiple coexisting attractors depending on the memristor initial value and the system parameter is studied, which shows the coexisting behaviors of point, period, chaos, and quasic-period. Secondly, with the system motion starting from a non-equilibrium point, the dynamics of extreme multistability in a wide initial value domain are easily conformed by new analytical methods. Furthermore, the simulation results indicate that some strange chaotic attractors like multi-wing type and multi-scroll type are observed when the observed signals are extended from voltage and current to power and energy, respectively. Specially, when different initial conditions are taken, the coexisting strange chaotic attractors between the power and energy signals are exhibited. Finally, the chaotic sequences of the new system are used for encrypting color image to protect image information security. The encryption performance is analyzed by statistic histogram, correlation, key spaces and key sensitivity. Simulation results show that the new memristive chaotic system has high security in color image encryption.

## I. INTRODUCTION

According to the combinations of circuit elements, memristor, a nonlinear resistor with memory, is predicted by Professor Chua.^{1} It was not until 2008 that the memristors were successfully implemented by the HP laboratory and its mathematical model was established.^{2} Since then, because of its memory and nonlinear properties, the memristor has a broad application prospects in the field of the image encryption,^{3,4} neural network^{5,6} and circuit design,^{7,8} and has become a new research hotspot in the field of nonlinear science. Circuits, composed of memristor, are prone to produce high frequency chaotic oscillation signals. As a controllable nanoscale device, the memristor can be used to advance the development of the traditional nonlinear field.

In recent years, many memristor-based chaotic circuits are implemented by replacing the nonlinear elements in classical chaotic circuits with the memristor, such as the memristor-based Lorenz system,^{9,10} the memristor-based Jerk circuit,^{11,12} the memristor-based Chua’s circuit,^{13,14} and so on. The memristor-based chaotic circuits not only have the dynamic behaviors of the classical systems, but also present some complex dynamic behaviors, such as state transition,^{15} coexisting attractor,^{16,17} extreme multistability^{18,19} and multi hidden attractors.^{20,21} At present, many kinds of memristor-based circuits have been reported and the corresponding dynamic characteristics have been analyzed.^{22–26} In Refs. 22 and 23, the dynamics of multistability in the memristor-based circuits are analyzed and it is found that the multiple attractors production extremely depends on the system parameters and initial values, especially the memristor initial condition. In addition, for coexisting many attractors depending on the system parameter, the analysis method is to draw the bifurcation diagram of the parameter with the different initial condition,^{24} and there seems to be little analysis of the relationship between parameter and initial values. For extreme multistability depending on initial conditions, the attraction basins is often used to described the distribution of attractors,^{25} but it is suitable for a narrow range of initial values.

In this work, the coexisting dynamical behaviors with the system motion starting from an equilibrium point are studied, which fully describes the relationship between parameter and the memristor initial value. When the system starts from the non-equilibrium point, one initial value is chosen as the bifurcation parameter, and its bifurcation diagrams with the memristor initial value in different interval is obtained, which can analyze extreme multistability in a broad range of initial value. Besides, the phenomenon of the coexisting strange attractors^{26} is found by extending to the observed signal. Furthermore, the application of chaotic sequence in color image encryption is implemented, and the application value of memristor chaotic circuits is well demonstrated.

The rest of the paper is organized as follows. In Sect. II, an active charge-controlled memristive chaotic system is proposed. Meanwhile, the stability of equilibrium point is analyzed. In Sect. III, numerical simulations for the coexisting behaviors depending on the memristor initial condition and the system parameter are performed. Besides, the coexisting many attractors’ behavior with the system initial value away from the equilibrium point is investigated through software Matlab and the coexistence phenomenon of strange chaotic attractors is studied in the terms of power and energy. In Sect. IV, memristive chaotic system is used for color image encryption and the safety performances analysis is performed. The conclusions are summarized in Sect. V.

## II. ACTIVE CHARGE-CONTROLLED MEMRISTIVE CHUA’S OSCILLATOR

### A. Mathematical model

A modified Chua’s oscillation circuit with an active charge-controlled memristor is shown in Fig. 1. The circuit consists of two capacitors, two inductors and one memristor that correspond to the state variables *v*_{1} *v*_{2} *i*_{L1} *i*_{L2} *q*. The memristor is a smooth nonlinear charge-controlled device and its relationship between charge and flux can be expressed as

where *m* and *n* are the memristive parameters. If *m*<0, the memristor is a positive device.

According to Kirchhoff’s voltage and current law, the state equations of a circuit in Fig. 1 are expressed as:

When *x* = *i*_{L1}, *y* = *i*_{L2}, *z* = *v*_{1}, *u* = *v*_{2}, *v* = *q*, *a* = 1/*L*_{1}, *b* = 1/*C*_{1}, *c* = 1/*C*_{2}, *L*_{2} = 1, *R* = 1, and the memristive model is expressed as *M*(*v*) = −*m* + *nv*^{2}, the formula (2) can be simplified as

where *x*,*y*,*z*,*u* and *v* are the state variables; *a*,*b*,*c* are the system parameters; *m*,*n* are the parameters of the charged-controlled memristive model.

When the parameters are given by *a*=9, *b*=30, *c*=17, *m*=-1.2, *n*=1.2 and initial conditions (10^{−6},0,0,0,0) are chosen, double-scroll chaotic attractor can be obtained as shown in Fig. 2. In this case, the corresponding Lyapunov exponents are calculated as *L*_{1} = 0.2571, *L*_{2} = 0.00, *L*_{3} = 0.00, *L*_{4} = −7.2514, *L*_{5} = −16.4492, and the Lyapunov dimension is *d*_{L} = 3.5746, which indicates the system is chaotic.

### B. The stability of the equilibrium point

Let $x\u0307=y\u0307=\u017c=u\u0307=v\u0307=0$ and the equilibrium set can be expressed as

where *l* can be any real constant value. According to the linearized system of Eq. (2), the Jacobian matrix on *A* is

When the parameters *c* and *l* are variable parameters, and other system parameters are given by *a* = 9, *b* = 30, *m* = −1.2, *n* = 1.2, the characteristic equation of the Jacobian matrix is

where *d*_{1} = 10.8*l*^{2} + 19.2, *d*_{2} = 324*l*^{2} + *c* − 24, *d*_{3} = (324 + 10.8*c*)*l*^{2} − 324 + 19.2*c*, *d*_{4} = 324*cl*^{2} − 54*c*.

According to the Routh-Hurwitz criterion, if the coefficients of the characteristic equation satisfy the Eq. (7), the equilibrium point is stable, leading to the occurrence of point attractor, whereas if any one of the four conditions is not satisfied, the equilibrium point is unstable, resulting in the emergence of periodic and chaotic attractors.

When the parameter *c* is increased from 15 to 20 and the parameter $l$ is changed from 0 to 1.2, the stability distribution of equilibrium points is shown in Fig. 3, where the green regions are unstable, whereas the cyan regions are stable. If *c* = 17 and *l* in different regions is selected, four nonzero eigenvalues of the system are calculated, as shown in Table I.

$l$ . | λ_{1}
. | λ_{2}
. | λ_{3}
. | λ_{4}
. | stability . |
---|---|---|---|---|---|

0.1 | 3.386 | -1.530+3.2619i | -1.530-3.262i | -19.633 | Unstable saddle point |

0.45 | -18.494 | -0.178+2.0434i | -0.178-2.043i | -2.537 | Stable focal point |

0.8 | 0.082+3.693i | 0.082-3.693i | -13.138+4.291i | -13.138-4.291i | Unstable saddle-focus point |

1 | -0.082+3.933i | -0.082-3.933i | -14.919+8.608i | -14.919-8.608i | Stable focal point |

$l$ . | λ_{1}
. | λ_{2}
. | λ_{3}
. | λ_{4}
. | stability . |
---|---|---|---|---|---|

0.1 | 3.386 | -1.530+3.2619i | -1.530-3.262i | -19.633 | Unstable saddle point |

0.45 | -18.494 | -0.178+2.0434i | -0.178-2.043i | -2.537 | Stable focal point |

0.8 | 0.082+3.693i | 0.082-3.693i | -13.138+4.291i | -13.138-4.291i | Unstable saddle-focus point |

1 | -0.082+3.933i | -0.082-3.933i | -14.919+8.608i | -14.919-8.608i | Stable focal point |

## III. OCCURRENCE OF MULTIPLE COEXISTING ATTRACTORS

In the following work, the dynamical behaviors for the system motion starting equilibrium point and non-equilibrium point are analyzed. When the system initial condition is an equilibrium point, and the system parameter *c* and the memristor initial value *v*(0) are adjustable, the system dynamics can well exhibit the coexisting infinitely many attractors’ behavior. When the state variable *x*(0) is not near to 0 and *v*(0) is also adjustable, the system initial condition will be a non-equilibrium point and extreme multistability will emerge in the proposed memristive system.

### A. Multiple coexisting attractors depending on *c* and $v$(0)

Firstly, the system parameters *a* = 9, *b* = 30, *c* = 17, *m* = −1.2, *n* = 1.2 are kept unchanged, and initial conditions are set as [10^{−6},0,0,0,*v*(0)] and [−10^{−6},0,0,0,*v*(0)] respectively. With *v*(0) increasing in the region [-1, 1], the bifurcation diagrams of the state variable *x* and Lyapunov exponent spectrum are shown in Fig. 4. Specially, the bifurcation behaviors of region I, II in Fig. 4(a) and the corresponding three largest Lyapunov exponents are shown in Table II.

Fig. 4(a) . | Region I . | Region II . | Fig. 4(b) . | Region I . | Region II . |
---|---|---|---|---|---|

Dynamic behavior (blue) | $Period\u2193Choas\u2193Quasi-period$ | $Chaos\u2193Period\u2193Point$ | (L_{1},L_{2},L_{3}) (b1) | $0,0,\u2212\u2193+,0,\u2212\u21930,0,\u2212$ | $+,0,\u2212\u21930,0,\u2212\u2193\u2212,\u2212,\u2212$ |

Dynamic behavior (red) | $Point\u2193Period\u2193choas$ | $Period\u2193Choas\u2193Quasi-period$ | (L_{1},L_{2},L_{3}) (b2) | $\u2212,\u2212,\u2212\u21930,0,\u2212\u2193+,0,\u2212$ | $0,0,\u2212\u2193+,0,\u2212\u21930,0,\u2212$ |

Fig. 4(a) . | Region I . | Region II . | Fig. 4(b) . | Region I . | Region II . |
---|---|---|---|---|---|

Dynamic behavior (blue) | $Period\u2193Choas\u2193Quasi-period$ | $Chaos\u2193Period\u2193Point$ | (L_{1},L_{2},L_{3}) (b1) | $0,0,\u2212\u2193+,0,\u2212\u21930,0,\u2212$ | $+,0,\u2212\u21930,0,\u2212\u2193\u2212,\u2212,\u2212$ |

Dynamic behavior (red) | $Point\u2193Period\u2193choas$ | $Period\u2193Choas\u2193Quasi-period$ | (L_{1},L_{2},L_{3}) (b2) | $\u2212,\u2212,\u2212\u21930,0,\u2212\u2193+,0,\u2212$ | $0,0,\u2212\u2193+,0,\u2212\u21930,0,\u2212$ |

When *v*(0) values in different regions are selected, the phase trajectories of the system display the coexisting single scroll chaotic attractor, periodic attractor and asymmetric chaotic attractor, as shown in Fig. 5(a) and (c). Fig. 5(a) shows the coexisting period-1 and single band chaotic attractor; Fig. 5(c) shows the coexisting asymmetric chaotic attractors. Meanwhile, the topological properties of coexisting phase trajectories are studied by using the Poincare map, as shown in Fig. 5(b) and (d).

Next, when *x*(0) = ±10^{−6}, *v*(0) = ±0.2, and *c* varies in the region of [16,19.5], the four nonzero eigenvalues are *λ*_{1} > 0,*λ*_{2,3} = *α*_{1} ± *jω*_{1},*λ*_{4} < 0 and *α*_{1} < 0, which corresponds to unstable region I in Fig. 3 and means that the system motion starts from the unstable saddle point. With *c* increasing in the region of [16,19.5], the bifurcation diagrams of the state variable *z* and Lyapunov exponent spectrum are shown in Fig. 6(a–d). When *v*(0) = ±0.63 and other conditions are kept unchanged, the four nonzero eigenvalues satisfy *λ*_{1,2} = *α*_{2} ± *jω*_{2},*λ*_{3,4} = *α*_{3} ± *jω*_{3} and *α*_{2} > 0,*α*_{3} < 0, which corresponds to unstable region II in Fig. 3 and indicates that the system trajectory starts from the unstable saddle-focus point. The corresponding bifurcation diagrams and Lyapunov exponent spectrum are shown in Fig. 6(e, f).

From Fig. 6, the four sets of different parameters *c* are selected and each parameter corresponds to six sets of different initial conditions. *IC*_{i}(*i* = 1,2,3,4) are in the unstable region I shown in Fig. 3; *IC*_{i}(*i* = 5,6) are in the unstable region II shown in Fig. 3, as shown in Table III. In the Table III, $Pij$ represents the *i*-th type of period-*j*; *C*_{i} represents the *i*-th type of asymmetric double scroll chaotic attractor; *S*_{i} represents the *i*-th type of single scroll chaotic attractor. The phase diagrams and Poincare maps of the coexisting attractors, listed in Table III, are shown in Fig. 7. Fig. 7(a) shows six types of the coexisting period-1 under *c* = 19.4; Fig. 7(b) shows the coexisting periodic limit cycles under *c* = 18.6; Fig. 7(c) shows the coexistence of periodic limit cycles and single scroll chaotic attractors under *c* = 17.7; Fig. 7(d) shows Poincare maps of the coexisting limit cycles and single scroll chaotic attractors under *c* = 17.7; Fig. 7(e) shows the coexistence of limit cycles and asymmetric double scroll chaotic attractors under *c* = 17.3; Fig. 7(f) shows Poincare maps of the coexisting limit cycles and under asymmetric double scroll chaotic attractors *c* = 17.3.

. | Parameter c
. | |||
---|---|---|---|---|

Initial condition IC_{i}(i = 1,2,3,4,5,6)
. | c = 19.4
. | c = 18.6
. | c = 17.7
. | c = 17.3
. |

IC_{1} = [10^{−6},0,0,0,0.2] | $P11$ (blue) | $P71$ (blue) | $P52$ (blue) | $P14$ (dark green) |

IC_{2} = [10^{−6},0,0,0,−0.2] | $P21$ (dark green) | $P12$ (dark green) | S_{1} (dark green) | $P24$ (blue) |

IC_{3} = [−10^{−6},0,0,0,0.2] | $P31$ (dark red) | $P22$ (dark red) | S_{2} (dark red) | $P14$ (dark red) |

IC_{4} = [−10^{−6},0,0,0,−0.2] | $P41$ (red) | $P81$ (red) | $P62$ (red) | $P24$ (red) |

IC_{5} = [10^{−6},0,0,0,0.63] | $P51$ (black) | $P32$ (black) | $P13$ (black) | C_{1} (black) |

IC_{6} = [10^{−6},0,0,0,−0.63] | $P61$ (green) | $P42$ (green) | $P23$ (green) | C_{2} (red) |

Phase diagrams | Fig. 7(a) | Fig. 7(b) | Fig. 7(c) | Fig. 7(e) |

. | Parameter c
. | |||
---|---|---|---|---|

Initial condition IC_{i}(i = 1,2,3,4,5,6)
. | c = 19.4
. | c = 18.6
. | c = 17.7
. | c = 17.3
. |

IC_{1} = [10^{−6},0,0,0,0.2] | $P11$ (blue) | $P71$ (blue) | $P52$ (blue) | $P14$ (dark green) |

IC_{2} = [10^{−6},0,0,0,−0.2] | $P21$ (dark green) | $P12$ (dark green) | S_{1} (dark green) | $P24$ (blue) |

IC_{3} = [−10^{−6},0,0,0,0.2] | $P31$ (dark red) | $P22$ (dark red) | S_{2} (dark red) | $P14$ (dark red) |

IC_{4} = [−10^{−6},0,0,0,−0.2] | $P41$ (red) | $P81$ (red) | $P62$ (red) | $P24$ (red) |

IC_{5} = [10^{−6},0,0,0,0.63] | $P51$ (black) | $P32$ (black) | $P13$ (black) | C_{1} (black) |

IC_{6} = [10^{−6},0,0,0,−0.63] | $P61$ (green) | $P42$ (green) | $P23$ (green) | C_{2} (red) |

Phase diagrams | Fig. 7(a) | Fig. 7(b) | Fig. 7(c) | Fig. 7(e) |

### B. Multiple coexisting attractors depending on *x*(0) and *v*(0)

Generally, the phenomenon of coexisting infinitely many attractors depending on double state variables *x*(0) and *v*(0) is described by attraction basins. Obviously, in order to present clearer dynamic behaviors, the range of parameter variation is relatively small. Therefore, we attempt to analyze extreme multistability in the wide parameter range by choosing *x*(0) as the bifurcation parameter, which means that the system initial condition is not an equilibrium point.

Through a large number of simulation analysis, it is found that the choice of *v*(0) in different intervals have influence on the bifurcation modes of the parameter *x*(0), as shown in Table IV. As seen from Table IV, when *v*(0) ∈ [1.87,1.87] and *v*(0) are chosen as 0,±0.5,±0.9,±1 respectively, the bifurcation diagrams of the state variable *y* with the increasing parameter *x*(0) are shown in Fig 8(a). From Fig 8(a), we can find that when *v*(0) takes any value within the region [-1.87, 1.87], the coexisting periodic, quasi-periodic and chaotic attractors can be obtained by changing the value of the state variable *x*(0). When *v*(0) ∈ [−2.20,−1.87) $\u222a$ (1.87,2.20] and *v*(0) are chosen as ±1.88,±1.9,±1.91 respectively, the bifurcation diagrams of the state variable *y* with the parameter *x*(0) increasing are shown in Fig 8(b). Compared with the bifurcation mode of Fig 8(a), the bifurcation mode of Fig 8(b) becomes discontinuous and the bifurcation diagram appears discontinuous periodic orbits. When *v*(0) ∈ [−3.15,−2.20) $\u222a$ (2.20,3.15] and *v*(0) are chosen as ±2.25,±2.27,±2.3 respectively, the bifurcation diagrams of the state variable *y* with the increasing *x*(0) are shown in Fig. 8(c). Fig. 8(c) shows that the bifurcation diagram is also discontinuous and the trajectories of the system change from stable fixed point to large 1-periodic orbit directly. When $v0>3.15$ and *v*(0) are chosen as ±3.17,±3.2,±3.22 respectively, the motion trajectories of the system only include stable fixed point and large 1-periodic orbit, as shown in Fig 8(d).

The region of the memristor initial condition v(0)
. | v(0) values
. | The bifurcation modes of the parameter x(0)
. |
---|---|---|

v(0) ∈ [−1.87,1.87] | v(0) = 0,±0.5,±0.9,±1 | Fig. 8(a) |

v(0) ∈ [−2.20,−1.87) $\u222a$ (1.87,2.20] | v(0) = ±1.88,±1.9,±1.91 | Fig. 8(b) |

v(0) ∈ [−3.15,−2.20) $\u222a$ (2.20,3.15] | v(0) = ±2.25,±2.27,±2.3 | Fig. 8(c) |

v(0) ∈ (−∞,−3.15) $\u222a$ (3.15,+∞) | v(0) = ±3.17,±3.2,±3.22 | Fig. 8(d) |

The region of the memristor initial condition v(0)
. | v(0) values
. | The bifurcation modes of the parameter x(0)
. |
---|---|---|

v(0) ∈ [−1.87,1.87] | v(0) = 0,±0.5,±0.9,±1 | Fig. 8(a) |

v(0) ∈ [−2.20,−1.87) $\u222a$ (1.87,2.20] | v(0) = ±1.88,±1.9,±1.91 | Fig. 8(b) |

v(0) ∈ [−3.15,−2.20) $\u222a$ (2.20,3.15] | v(0) = ±2.25,±2.27,±2.3 | Fig. 8(c) |

v(0) ∈ (−∞,−3.15) $\u222a$ (3.15,+∞) | v(0) = ±3.17,±3.2,±3.22 | Fig. 8(d) |

### C. Coexisting strange attractors

Usually observed chaotic attractors describe the relationship of voltage cross the capacitor (*v*), current in the inductance branch (*i*), charge (*q*) and magnetic flux (*φ*) of the memristor. If the observed state variables are generalized to power (*P*) and energy (*W*) signals, it is found that the folding and extension of attractors are more complex. when the system parameters are gave by *a* = 9, *b* = 30, *c* = 17, *m* = −1.2, *n* = 1.2 and initial conditions are (10^{−6},0,0,0,0), numerical simulation for the relationship of different state variables are performed, as shown in Fig. 9. Fig. 9(a) shows a four-wing chaotic attractor, which reflects the relationship between voltage (*v*_{1}) cross the capacitor *C*_{1} and energy (*W*(*L*_{1})) in the inductor *L*_{1}; Fig. 9(b) displays a four-scroll chaotic attractor, which describes the characteristic between power (*P*(*M*(*q*))) of the memristor *M*(*q*); Fig. 9(c) exhibits a four-wing and double-scroll chaotic attractor, which implies the relationship between power (*P*(*M*(*q*))) of the memristor *M*(*q*) and voltage (*v*_{1}) cross the capacitor *C*_{1}; Fig. 9(d) reveals a double-wing and double-scroll chaotic attractor, which expresses the relationship of power (*P*(*M*(*q*))) of the memristor *M*(*q*) and energy (*W*(*L*_{2})) in the inductor *L*_{2}.

In addition, the power (*P*), energy (*W*), voltage (*v*) and charge (*q*) are choose as the independent signal to analyze the coexisting dynamical behaviors. When the system initial values are changed, the symmetry of strange chaotic attractor will be destroyed, and it is found that the system motion has a large number of coexisting behaviors, as shown in Fig. 10. In Fig. 10(a), with the initial conditions *IC*_{1} = (10^{−6},0,1.3,0,0) and *IC*_{2} = (10^{−6},0,−1.3,0,0) respectively, the system emerges the coexisting double-wing chaotic attractors in *v*_{1} − *W*(*L*_{1}) plane. Fig. 10(b) displays the coexisting double-scroll chaotic attractors in *q* − *W*(*v*_{2}) plane for the initial conditions *IC*_{1} and *IC*_{2}. In Fig. 10(c), for initial conditions *IC*_{3} = (10^{−6},0,0.5,0,0) and *IC*_{4} = (10^{−6},0,−0.5,0,0), there are the coexisting double-wing and single-scroll chaotic attractors in *P*(*M*(*q*)) − *v*_{1} plane. Fig. 10(d) exhibits the coexisting single-wing and single-scroll chaotic attractors in *P*(*M*(*q*)) − *v*_{1} plane with the initial conditions *IC*_{3} and *IC*_{4} respectively.

## IV. THE APPLICATION ON IMAGE ENCRYPTION

### A. Analysis of image encryption algorithm

In this section, we adopt the method of combining pixel displacement and pixel location scrambling, and design an image encryption scheme with chaotic sequence, as shown in Fig 11. The scheme can be outlined as follows:

Step 1: we obtain RGB pixel sequences: $CM\xd7NR$ $CM\xd7NG$ $CM\xd7NB$ from the

*M*×*N*digital color image (*M*is the row pixel,*N*is the column pixel). Two chaotic sequences with length*M*×*N*are selected and processed to obtain new sequences*X*_{1}*X*_{2}:

Step 2: we combine sequence

*X*_{1}with pixel sequences $CM\xd7NR$ $CM\xd7NG$ $CM\xd7NB$ by XOR, and obtain new sequences $X1\u2032$ $X2\u2032$ $X3\u2032$:

Step 3: A new sequence is given by

*T*= {*t*_{1},*t*_{2}⋯,*t*_{i},⋯,*t*_{M×N}} = {1,2⋯,*M*×*N*} and the element*t*_{i}of sequence*T*is handled as follows: we take the element*x*_{i}of sequence*X*_{2}and exchange the*i*-th element of sequence*T*with the*x*_{i}−*th*element of sequence*T*, and then move the element*t*_{i}backward*x*_{i}bit to obtain a new sequence $T\u2032$.Step 4: the new sequence $T\u2032$ is used as address mapping table to rearrange each element address of the sequence $X1\u2032$ $X2\u2032$ $X3\u2032$ and then we can obtain encrypted RGB pixel matrixs $X1\u2033$ $X2\u2033$ $X3\u2033$ which can be synthesized to encrypted color image. The decryption process is the reverse process of the encryption, which we omit here.

### B. Experimental simulation and performance analysis

In the following simulation, we select the standard Lena color digital image, whose size is 256 × 256. In addition, the system parameters of memristor chaotic system are set as *a*=9, *b*=30, *c*=17, *m*=-1.2, *n*=1.2, and the system initial conditions, with values as [0.1,0.1,0,0,0.4], [0,0.1,0,0,0.3], [0.1,0,0,0,0.4] are chosen as the algorithm keys of RGB image respectively. The simulation results are shown in Fig 12. The cipher RGB images completely hide useful information, which means that the algorithm can make good use of chaotic sequence to encrypt the color image. In order to present the validity of algorithm, we also adopt other analysis methods including image histogram analysis, pixel correlation analysis, and key sensitivity analysis.

#### 1. Statistical analysis

Image histogram is one of the most important statistical methods. In this paper, Lena.bmp color images R G B are analyzed statistically, as shown in Fig 13. Fig 13(a–c) show plain RGB image histograms, whose distribution is extremely uneven; Fig 13(d–f) show cipher RGB image histogram with uniform distribution. These indicate that the encryption algorithm can effectively diffuse plain information into random cipher, and resist statistical attacks.

#### 2. Correlation analysis

Correlation analysis is usually applied to study the scattering of encryption algorithm, and it is one of the important performance indexes to evaluate image encryption performance. By analyzing the adjacent pixels distribution of plain and cipher image, we can understand the effectiveness of memristor chaotic encryption algorithm. Besides, the correlation of adjacent pixels can be calculated by the following formula:

where, *x* and *y* are the pixel values of two adjacent pixels respectively; *ρ*_{x,y} indicates the correlation coefficient of adjacent image pixels.

Fig. 14 (a–c) presents the adjacent pixel correlation of plain image R G B in the horizontal direction, while Fig. 14 (d–f) shows the adjacent pixel correlation of cipher image R G B in the horizontal direction. It is obvious that the correlation between the pixels of the plain image is linear, while the correlation between the pixels of the cipher image is random. As can be seen from Table V, the pixel correlation coefficient of plain image RGB in the horizontal, vertical or diagonal directions is close to 1, and the pixel correlation coefficient of encrypted image RGB in horizontal, vertical and diagonal directions is very close to 0. The encrypted image has no features of the original image, which means it can effectively resist plain-text attacks.

. | Plain . | Cipher . | ||||
---|---|---|---|---|---|---|

Direction . | R . | G . | B . | R . | G . | B . |

horizontal | 0.9571 | 0.9457 | 0.9291 | 0.0029 | -0.0098 | 0.0056 |

vertical | 0.9624 | 0.9481 | 0.9336 | -0.0118 | 0.0036 | -0.0183 |

diagonal | 0.9248 | 0.8953 | 0.8512 | 0.1380 | 0.2343 | 0.1939 |

. | Plain . | Cipher . | ||||
---|---|---|---|---|---|---|

Direction . | R . | G . | B . | R . | G . | B . |

horizontal | 0.9571 | 0.9457 | 0.9291 | 0.0029 | -0.0098 | 0.0056 |

vertical | 0.9624 | 0.9481 | 0.9336 | -0.0118 | 0.0036 | -0.0183 |

diagonal | 0.9248 | 0.8953 | 0.8512 | 0.1380 | 0.2343 | 0.1939 |

#### 3. Secret key space and sensitivity analysis

The system parameters: *a b c m n* and the initial values: *x*_{0} *y*_{0} *z*_{0} *u*_{0} *v*_{0} are chosen as secret keys. The key space is up to 10^{150}, which is much larger than 2^{100}, which means that the exhaustive attacks will lose its effect. For the key sensitivity, we change system parameter *a* and initial value *x*_{0} by 10^{−15}, respectively, and the remaining key parameters are unchanged. The decrypted image is shown in Fig 15. Other secret key parameters have the same sensitivity, which indicates that the algorithm is highly sensitive to the secret keys.

## V. CONCLUSION

By replacing the nonlinear resistance in Chua’s circuit with a charge-controlled memristor model, a fifth-order memristive chaotic system is presented in the paper. One feature of the proposed memristive chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. Numerical simulations for coexisting attractors are performed, which show the coexisting behaviors of point, period, chaos, and quasic-period. More interestingly, the first state variable of newly proposed memristive system has different bifurcation structures with the memristor initial condition in different regions and its multistability closely relies on memristor initial condition and the first state variable, which reveals that the complex dynamical behaviors with the system motion start from the non-equilibrium point. Another feature is that some strange chaotic attractors like mixed type and superimposed type are obtained by observing the power and energy signal, and the dynamic behaviors of coexisting strange attractors are exhibited by phase diagrams. At last, color image encryption based on the new chaotic system is studied and safety performance analysis is performed, including key spaces, key sensibility and anti-attack ability. Results show that the new system can be used to encrypt color images in a very secure way.

## ACKNOWLEDGMENTS

This work is supported by the National Nature Science Foundation of China under Grant No.51475246 and the Natural Science Foundation of Jiangsu Province of China under Grant No.Bk20131402.