#### Abstract

In this work, the interesting dynamics of coupled nonlinear memristor-based oscillators in ring configuration are explored. The mathematical models are derived to describe the possible cases of employing identical or different nonlinearities. Analytical and numerical techniques, involving perturbation methods, normal forms, phase portraits, and Lyapunov exponents are used to investigate various types of dynamical behaviors along with their stability regions in parameters space. The effects of time-delayed coupling on the proposed system are numerically studied. It is demonstrated that the coupled oscillators show rich dynamics including periodic orbits, quasiperiodicity, two-dimensional, and three-dimensional tori.

#### 1. Introduction

Many applications in engineering, physics, chemistry, biology, and economy are modeled using nonlinear differential equations (DEs) where their exact forms of solutions are very difficult to obtain in general or they do not provide tools to understand the underlying dynamics. However, dynamical systems methods provide powerful techniques to study the qualitative behaviors of mathematical model solution rather than quantitative ones [1].

One of the interesting problems in nonlinear theory is to study the dynamics of systems involving coupled nonlinear oscillators. This problem appears in many applications in biology [2], neuroscience [3–6], chemistry [7, 8], and physics [9–11]. Researchers in the field of dynamical systems are interested in studying the stability of coupled oscillators systems with nonlinear properties since 1950s till now [12–15].

In 1970, Chua introduced the fourth electronic element called the memristor [16]. It is a two-terminal element that relates the magnetic flux between the terminals with the electric charge that passes through it [16, 17]. Studies on the dynamical properties, i.e., stability and bifurcation of the memristor-based oscillator circuit, have received increasing interest [18–21]. The main reason is that coupled memristor-based oscillatory circuits appear in many applications such as filtering, energy storage, energy transmission, isolation impedance transformation, learning networks that require a synapse-like function, secure communications, and image stabilization [22–31].

In [32], the stability of three nonlinear identical memristor-based oscillators coupled by inductors in ladder circuit configuration was studied. In this paper, we examine a more generalized case by considering coupled three memristor-based oscillators in ring configuration where nonidentical memristors are allowed. Two cases are studied where both identical and different nonlinearities are considered. The memristor used is of a flux-controlled type that has memductance function representing the flux-dependent rate of change of charge The two cases studied are according to the definition of . In the first case the memristors are described by three third-order polynomials, whereas in the second case one memristor is described by third-order polynomial while the remaining memristors are characterized by fourth-order polynomials.

Time delays play an important role in mathematical modeling of real systems and studying synchronization such that their effects cannot be neglected in real life; see, for example, [33–35]. Therefore, numerical investigations of time-delayed coupling influences on coupled oscillators dynamics are carried out in this work. The rest of the paper is organized as follows: In Section 2, mathematical models of all cases studied in this work are presented. In Section 3, the multiple scales method is applied to acquire the normal form for each system. In Section 4, various types of dynamical behaviors that can be extracted from normal forms in addition to their stability analysis are obtained. In Section 5, numerical simulations are carried out to verify theoretical results. The general discussion and main conclusion of this work are presented in Section 6.

#### 2. Ring Model Equations

In this section, we derive the fundamental equations describing the three ring coupled oscillators circuit. Figure 1 shows the memristor-based oscillator circuit we adopt in this work. Applying Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) to the circuit in Figure 2, the model can be described as follows:We consider two cases of system (2) according to the equations describing memristors.

##### 2.1. Case 1

In this case, the nonlinearities of the three memristors are described as follows:By integrating both sides of each equation of system (2) between times and while using (3) and the relationswe getBy differentiating (6)–(8) and using (9)–(14), we obtain the following equations that depict our intended case:

##### 2.2. Case 2

In this case, the nonlinearities of the three memristors are different and described as follows:Using the same procedure as in Section 2.1, we obtain

#### 3. Normal Forms

The purpose of this section is to acquire the normal forms corresponding to each case of ring model derived in previous section. The powerful perturbation techniques are used to achieve this goal. First, systems (15) and (17) are written in perturbed form; then each system is reduced to an equivalent first-order system [36].

System (15) can be written in perturbed form as follows:And system (17) can be written in perturbed form as follows:By using the substitutionswhere is the complex conjugate of , systems (18) and (19) are reduced to the more simplified first-order systems of DEs. More specifically, system (18) is reduced to the following system:And system (19) is reduced to the following system:The normal forms corresponding to systems (21) and (22) are obtained by applying the method of multiple scales [36].

##### 3.1. The Normal Form of Case 1 of Ring Model

Using the substitutionsin system (21) and equating coefficients of equal powers of , we obtain

(**Order **)

(**Order **)

(**Order **)Solving system (24) of DEs we getBy substituting these solutions in (25) we get the conditions needed to remove the secular terms in the following forms:Using (28) to solve system (25) and substituting from the results in (26), we get the following conditions to remove the secular termsSubstituting from (28) and (29) in , where and 3, we obtain the normal form of system (21) which can be expressed in the following form:

##### 3.2. The Normal Form of Case 2 of Ring Model

Using the same procedure applied in Section 3.1, the following conditions are obtained to remove the secular terms of order and the conditions needed to remove the secular terms of order are given bySo, we get the following normal form:

#### 4. Stability Analysis of Equilibrium Solutions

In this section, the stability analysis of the equilibrium solutions is carried out using the normal forms obtained in (30) and (33).

##### 4.1. Stability Analysis of Case 1 of Ring Model

The Jacobian matrix of system (30) is expressed asSystem (30) has the following equilibrium solutions.

###### 4.1.1. Equilibrium Point

The following equilibrium solution,is corresponding to an equilibrium point at origin. Evaluating the eigenvalues of Jacobian matrix (34) at equilibrium point (35), we get the stability region of this point in parameters space as follows:

###### 4.1.2. Periodic Solution I

The second equilibrium solution,corresponds to a periodic solution in the phase space of case of ladder model. The second order approximate solution of this periodic orbit has an amplitude given by and an angular frequency computed asEvaluating the eigenvalues of Jacobian matrix (34) at periodic solution I (37), we get its stability region in parameters space as follows:

###### 4.1.3. Periodic Solution II

The third equilibrium solution,represents a periodic solution in the phase space of case of ring model. The second order approximate solution of this periodic orbit has an amplitude given by and an angular frequency computed asEvaluating the eigenvalues of Jacobian matrix (34) at periodic solution II (40), we get its stability region in parameters space as follows:

###### 4.1.4. Periodic Solution III

The fourth equilibrium solution,corresponds to a periodic solution in the phase space of case of ring model. The second order approximate solution of this periodic orbit has an amplitude given by and an angular frequency computed asEvaluating the eigenvalues of Jacobian matrix (34) at periodic solution III (43), we get its stability region in parameters space as follows:

###### 4.1.5. 2D Torus I

The fifth equilibrium solution,corresponds to a 2D torus in the phase space of case of ring model. The second order approximate solution of this 2D torus has two fundamental angular frequencies given by (44) and (41). Evaluating the eigenvalues of Jacobian matrix (34) at 2D torus I (46), we get the stability region in parameters space as follows:

###### 4.1.6. 2D Torus II

The sixth equilibrium solution,corresponds to a 2D torus in the phase space of case of ring model. The second order approximate solution of this 2D torus has two fundamental angular frequencies given by (44) and (38). Evaluating the eigenvalues of Jacobian matrix (34) at 2D torus II (48), we get the stability region in parameters space as follows:

###### 4.1.7. 2D Torus III

The seventh equilibrium solution,corresponds to a 2D torus in the phase space of case of ring model. The second order approximate solution of this 2D torus has two fundamental angular frequencies given by (41) and (38). Evaluating the eigenvalues of Jacobian matrix (34) at 2D torus III (50), we get the stability region in parameters space as follows:

###### 4.1.8. 3D Torus

The eighth equilibrium solution,corresponds to a 3D torus in the phase space of case of ring model. The second order approximate solution of this 3D torus has three fundamental angular frequencies given by (44), (41), and (38). Evaluating the eigenvalues of Jacobian matrix (34) at 3D torus (52), we get the stability region in parameters space as follows:

##### 4.2. Stability Analysis of Case 2 of Ring Model

The Jacobian matrix of system (33) is given bySystem (33) has the following equilibrium solutions.

###### 4.2.1. Equilibrium Point

The equilibrium solutionconstitutes an equilibrium point at origin for case of ring model. Evaluating the eigenvalues of Jacobian matrix (54) at equilibrium point (55), we get the following region of stability in parameters space:

###### 4.2.2. Periodic Solution I

The equilibrium solution,constitutes a periodic solution in the phase space of case of ring model. The second order approximate solution of this periodic orbit has an amplitude and an angular frequency given asEvaluating the eigenvalues of Jacobian matrix (54) at periodic solution I (57), we get its stability region in parameters space as follows:

###### 4.2.3. Periodic Solution II

The equilibrium solution,constitutes a periodic solution in case of ring model phase space. The second order approximate solution of this periodic orbit has an amplitude and an angular frequency given asEvaluating the eigenvalues of Jacobian matrix (54) at periodic solution II (60), we get its stability region in parametric space as follows:

###### 4.2.4. 2D Torus

The equilibrium solution,corresponds to a 2D torus in the phase space of case of ring model. The second order approximate solution of this 2D torus has two fundamental angular frequencies given by (61) and (58). Evaluating the eigenvalues of Jacobian matrix (54) at 2D torus (63), we get the following stability region in parametric space:

#### 5. Numerical Simulations

In this section, numerical simulations are carried out to verify the theoretical results obtained in Section 4.

##### 5.1. Numerical Simulation of Case 1 of Ring Model

Some numerical examples of system (18) are given to address the following scenarios. First, when parameters , , and have negative values, it is found that a stable equilibrium point at the origin exists as shown in Figure 3. This scenario always occurs for values of parameters satisfying conditions (36). Second, a stable periodic orbit appears in system dynamics when parameters and have negative values while is positive. The equilibrium point at the origin loses its stability and this scenario arises for values of parameters satisfying conditions in (39). Periodic orbits are still observed when parameters , , and satisfy conditions given in (42) and (45). Figure 4 shows the stable periodic orbit that emerges at , , and . In third scenario, parameters and are positive while is negative. In this case, a 2D torus appears as shown in Figure 5. This agrees with previously obtained stability conditions in parameter space given by (47). The Lyapunov exponents spectrum plot associated with the 2D torus case is shown in Figure 6. Note that 2D tori can also be spotted in system dynamics when conditions (49) and (51) are satisfied. Finally, the parameters , , and are taken positive in numerical simulations. In particular, Figure 7 clarifies the occurrence of 3D torus in phase space of the system at , , and . This case is corresponding to the conditions given in (53). The associated Lyapunov exponents spectrum is plotted and shown in Figure 8.

**(a)**--

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**(d)**-##### 5.2. Numerical Simulation of Case 2 of Ring Model

The numerical examples of system (19) are shown in Figures 9–12 to illustrate the following scenarios. First, a stable equilibrium point at origin occurs when parameters , , and are all negative as shown in Figure 9. Setting the values of parameters to satisfy conditions (56) leads to this scenario. Second, when parameters and are negative while is positive, a stable periodic orbit is obtained. The stable equilibrium point at origin becomes unstable and this scenario arises for the values of parameters conditions in (59). A stable periodic orbit is shown in Figure 10 when , , and . Periodic orbits also occur when parameters , , and satisfy conditions in (62). Finally, a 2D torus appears in system dynamics when parameters and are positive while is negative. This is clarified in Figure 11 where , , and . The values of parameters , , and in this example satisfy the conditions in (64). The associated Lyapunov exponents spectrum is plotted and shown in Figure 12.

**(a)**--

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**(d)**-##### 5.3. Effects of Time-Delayed Coupling

In this part, the influences of time-delayed coupling on the aforementioned numerical examples are numerically investigated. For case of ring model the mathematical model can be rewritten in the following form: and denotes constant time delay value. It is observed that for the values of parameters used in Figure 13 the equilibrium point loses its stability when time delay value increases over a certain value (approximately 0.4) and a periodic orbit is induced.

**(a)**

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**(c)**

However, the effects of time delay coupling on other dynamics of system are almost negligible as shown in Figure 14.

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