The chaotic mechanisms in some jerk systems

: In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which ε is a small parameter, and a , b , c , d are some other parameters. For ε = 0, the system is Z 2 symmetric. For ε (cid:44) 0, the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor’s theorem and Hassard’s formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter a , the mechanisms are identified by means of bifurcation diagrams. Cases A and B are Z 2 symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincar´e sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system


Introduction
Chaos theory is a very important domain in mathematics, with a variety of applications.It has been applied to disease control and prevention [1], mechanics [2], biology [3], secure communications [4], and many other disciplines.Even small changes in a chaotic system's initial conditions can result in complex and unpredictable behavior.In the past few decades, many systems with chaotic attractors have been discovered, such as the Lorenz system [5], the Rössler system [6], the Chen system [7], the Yang systems [8] and the Sprott systems [9].
The study of chaotic flows with various equilibrium types is an interesting topic; see [10,11] and the references therein.In [10], some chaotic systems with a unique equilibrium were presented, admitting various equilibrium types.In the review paper [11], the authors classified chaotic systems with special equilibrium properties into eight groups: systems with no equilibrium (NE), systems with stable equilibrium (SE), systems with line equilibrium (LE), systems with curve equilibrium (CE), systems with plane equilibrium (PE), systems with surface equilibrium (ES), systems with unstable equilibrium (US) and systems that fall into more than one category (Chameleon systems).
Since the seminal work of Leonov et al. [12], the attractors in dynamical systems have been categorized into two categories: self-excited and hidden.A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium.Conversely, an attractor is called hidden if its basin of attraction does not intersect with a small neighborhood of any equilibrium point.The study of systems with hidden attractors has become a popular trend because hidden attractors play a significant role in many theoretical problems and engineering applications.Hidden attractors can be generated by a system in the following four types: Type 1, systems without equilibrium points [13]; Type 2, systems with only stable equilibrium points [14]; Type 3, systems with an infinite number of equilibrium points [15,16]; Type 4, systems with coexisting self-excited attractors [17].
It is well known that the presence of hidden attractors is connected with multistability [18][19][20][21].For a compass-gait robot model with multistability, Zavareh et al. [21] found that the strange attractors were hidden by the help of a basin of attractions.The concept of perpetual points refers to points where a system's acceleration becomes zero, but its velocity remains nonzero.In some situations, these points may be useful for locating attractors, comprehending dynamics and determining whether the system is conservative or not.For the definition, existence, properties and applications of perpetual points; see [22][23][24][25].Nowadays, the transmission of data with security is an important topic.Encryption with chaos is based on the ability of some dynamic systems to produce a sequence of bits that appear to be random in nature.Recently, a novel chaotic system was presented in [26] with a unique feature of crossing inside and outside of a cylinder repeatedly.Using a novel chaotic system with a hidden attractor, an encryption algorithm was designed in [27] for image encryption.In a given system, multistability refers to the superposition of several different attractors with the same set of parameters, starting from different initial conditions.A particular feature of multistable systems is their sensitivity to initial conditions.This type of system has a wide variety of states, giving it a great deal of flexibility to meet the needs of different applications [28].In recent years, considerable attention has been attracted to chaotic systems with multistability.The phenomenon of multistability has been observed in various fields of science, such as secure communications [4], image processing [29], neural systems [30] and economic systems [31].Extreme multistability means the coexistence of infinitely many attractors; see [32].Megastability refers to the coexistence of nested infinite attractors; see [33].
As the special case of 3D flows, jerk systems are very attractive due to simple form and theoretical and practical importance.In the recent years, although great efforts have been paid to them, the forming mechanisms of attractors are still far from complete.In fact, one of main problems of bifurcation theory is to understand the transition to chaos.
In physics, jerk is the third derivative of position with respect to time.Therefore, differential equations of the form ...
are called jerk equations, where the dot denotes differentiation with respect to time t.The term "jerk" comes from the fact that in a mechanical system, in which x is the displacement, ẋ is the velocity, and ẍ is the acceleration, the quantity ...
x is called the "jerk".The jerk function J(x, ẋ, ẍ) corresponds to a force F per unit mass that satisfies dF dt = J(x, ẋ, ẍ).
Letting y = ẋ, z = ẍ, Eq (1.1) can be transformed into the jerk system: Many chaotic jerk systems can be found with various forms of J(x, y, z), such as piecewise nonlinear functions, trigonometric functions, absolute value functions or power functions; for more information see [9].For polynomial jerk systems, the regular and chaotic behavior was studied in [34][35][36][37].In these studies, the qualitative features of equilibria played an important role in determining the complex behavior of the system.The differential system in the form is called a hyperjerk sytem, where n > 3. Due to their simplicity and complex dynamic properties, such systems have received considerable attention.For some hyperjerk systems with n = 4, 5, there are many studies on bifurcation analysis, numerical analysis, synchronizations and circuit implementations in [38][39][40][41][42][43] and references therein.As an exciting field of chaos, hidden attractors of 4D hyperjerk systems were investigated in [40,42].Joshi and Ranjan [44] introduced a jerk equation, ...
Rajagopal et al. [45] found various dynamical behaviors: periodic attractors, one-scroll chaotic attractors and coexistence between chaotic and periodic attractors.Volos et al. [46] presented a new jerk equation, ... x + x + b ẍ + a sinh( ẋ) = 0, which exhibits many interesting chaotic phenomena, such as coexisting attractors and antimonotonicity.Kengne et al. [47] considered the dynamics of a novel jerk system, which can be rewritten as Using bifurcation analysis, it was found that the system exhibits a period-doubling route to chaos, symmetry recovering crises and multistability.Kengne et al. [48] presented a jerk system, which can be rewritten as Some unusual and striking nonlinear phenomena were observed in the system: coexisting bifurcation branches, hysteretic dynamics, coexisting asymmetric bubbles, critical transitions and multiple coexisting asymmetric attractors.Li et al. [49] considered a cubic jerk system, for which the rich dynamical behaviors were observed, such as period-doubling bifurcation and reverse period-doubling bifurcation routes to chaos, crisis, multiple symmetric coexisting attractors and antimonotonicity.There are a few jerk systems in the literature which exhibit hidden chaotic attractors.Some of them are listed as follows.For hidden chaotic systems with one stable equilibrium, see cases SE1-SE6 in [50].More examples can be found in [51][52][53].For hidden chaotic systems without equilibrium, see cases NE3, NE6, NE14, NE15 in [54].Some other examples can be found in [55,56].
This paper is organized as follows.In the next section, four jerk systems with hyperbolic sine nonlinearity are presented, in which Cases A and C are Z 2 symmetric, and Cases B and D are asymmetric.In Sections 3 and 4, two mechanisms for chaos are studied via bifurcation diagrams, Poincaré cross sections, phase portraits and Lyapunov exponents.Section 5 is devoted to the local bifurcation analysis for a general jerk system, which includes Cases A and B. In Section 6, the electronic circuit for a chaotic jerk system is designed and simulated through PSIM 9.0.3.The outputs show good agreement with the numerical outputs.Finally, some concluding remarks are drawn in the last section.

The jerk systems
Consider the following equation: where 0 2) since a, b, c, d and ε are independent parameters.In fact, by setting Letting y = ẋ, z = ẍ, Eq (2.1) can be transformed into the following jerk system: (2.2) The system is elegant with a nonlinearity in the third equation, which has great influence on the longterm behavior.

The symmetric case with
It is Z 2 symmetric with respect to the origin, i.e., if (x(t), y(t), z(t)) is a solution of this system, then (−x(t), −y(t), −z(t)) is also a solution.Under some appropriate conditions, the symmetry can produce a single symmetric attractor or a symmetric pair of attractors.
The system is conservative if c = 0, the system is explosive if c < 0, and the system is dissipative if c > 0. From now on, we assume that b > 0, c > 0, d 0.
(2.4) Thus, the system is dissipative with a negative divergence ∇ V = −c, i.e., the flow of the system uniformly contracts the volumes of the state space, and all orbits of the system are eventually confined to a specific subset of zero volume.
To understand the chaotic mechanisms of system (2.3), the following two special cases are mainly considered in the next two sections: Case A : and Case B : (2.6) It should be remarked that, although these two systems are very special, the underlying mechanisms can also be found in many other jerk systems, such as the following polynomial jerk systems:

The asymmetric case with
which is an asymmetric jerk system.
To understand the chaotic mechanisms of system (2.7), the following two special cases are also considered in the next two sections: and Case D : (2.9)

Mechanism I: forward period-doubling route to chaos
This section is about the dynamical analysis and bifurcation analysis of Cases A and C. For these systems, the bifurcation diagrams will be plotted, depicting the local maxima of x(t) versus the parameter a.It would be interesting to see that the underlying mechanisms are the same: forward period-doubling routes to chaos.

Case A
With the initial conditions S 1,2 , the bifurcation diagram of system (2.5) with respect to the parameter a is plotted in Figure 1.Here, the blue branches correspond to S 1 , and the red branches correspond to S 2 .
Due to Z 2 symmetry, we only describe the long term behaviors of the trajectory starting from the point S 1 , corresponding to the blue branches in Figure 1.When the parameter a varies from 1.1 to 1.67, the system displays a stable equilibrium up to a = 1.2, where the supercritical Hopf bifurcation triggers a period-1 limit cycle.For 1.2 < a < 1.4, the amplitude of the cycle grows with the increase of a.For 1.4 < a < 1.58, the fluctuations of the amplitude with jumps can be observed.At about a = 1.58, the cycle exhibits the first period-doubling bifurcation.With further increase in parameter a, the system shows a period-doubling route to chaotic oscillations.For 1.618 ≤ a ≤ 1.67, the system enters a chaotic state.Furthermore, the corresponding Lyapunov exponents versus a are shown in Figure 2, in which the initial conditions correspond to S 1 .In Figure 2, stable equilibrium corresponds to the largest Lyapunov exponent L 1 that is less than zero, periodic dynamics correspond to L 1 that is equal to zero, and chaotic behavior corresponds to L 1 that is greater than zero.The bifurcation diagram and the Lyapunov exponent spectrum are in very good agreement.In fact, a subcritical pitchfork bifurcation also occurs at the origin for a = 1, which is out of the scope of this section.For more details about the local bifurcations, see Section 5.
For a = 1.3, 1.5, 1.605, the 2D views of the periodic attractors are shown in Figure 3.
The coexistence of chaotic attractors can be found at a = 1.618; see Figure 4.
A fully developed one-scroll chaotic attractor can be found at a = 1.67, which is shown in Figure 5.
To illustrate the folding and stretching structure of the attractor, cross sections (two-sided Poincaré sections) are plotted in Figure 6.If the parameter a is further increased, we see that a boundary crisis occurs at a = 1.6777.
The 3D chaotic attractors of the system for a = 1.618 and a = 1.67 are plotted in Figure 7.For the latter chaotic attractor, the Lyapunov exponents are 0.1071, 0, -0.3071, and thus the Kaplan-Yorker dimension is D KY = 2.3487.
With the other initial conditions S 3 -S 6 and a ∈ [1.58, 1.59], the bifurcation diagram of system (2.5) is plotted in Figure 8, which illustrates the domain of the coexistence of multiple periodic attractors.Here, the blue branches correspond to S 3 , the red branches correspond to S 4 , the green branches correspond to S 5 , and the black branches correspond to S 6 .For a = 1.584, two period-2 and two period-3 limit cycles can be observed in Figure 8.

Case C
Let us consider system (2.8), which is denoted as Case C. It is obtained from system (2.5) by adding a small constant term 0.01 in the jerk function.In comparison to system (2.5), it is natural for system (2.8) to have some similar dynamical properties, such as the chaotic mechanism and multistability.For more details, please see Figure 9.With the initial conditions S 3 -S 6 and a ∈ [1.58, 1.656], the bifurcation diagram of system (2.8) is shown in Figure 10.Here, the blue branches correspond to S 3 , the red branches correspond to S 4 , the green branches correspond to S 5 , and the black branches correspond to S 6 .For a = 1.58, period-1 (red), period-2 (green) and period-3 (black) attractors coexist.For a = 1.627, two coexisting chaotic attractors (green and black) can be observed.

Mechanism II: backward period-doubling route to chaos
This section is about the dynamical analysis and bifurcation analysis of Cases B and D. For these systems, the bifurcation diagrams will be plotted, depicting the local maxima of x(t) versus the parameter a.It would be interesting to see that the underlying mechanisms are the same: backward period-doubling routes to chaos.

Case B
The bifurcation diagram of system (2.6) is plotted in Figure 11, which depicts the local maxima of x(t) with respect to the parameter a.Here, the two initial conditions are S 1,2 : (x, y, z) = (0, 0, ±0.1).The blue curves correspond to S 1 , while red ones correspond to S 2 .
Due to Z 2 symmetry, we only describe the long term behaviors of the trajectory starting from the point S 1 , corresponding to the blue branches in Figure 11.From the bifurcation diagram, we can see that the system can exhibit a point attractor ( −1.09 ≤ a ≤ −0.83), a pitchfork bifurcation (occurs at the origin when a = −1), supercritical Hopf bifurcations (occur at the two nontrivial equilibria for a ≈ −1.1 ), periodic attractors (−1.33 < a < −1.11) and a backward period-doubling cascade from chaotic to periodic dynamics.These observations are in good agreement with the Lyapunov exponent spectrum in Figure 12.For more details about the local bifurcations, see Section 5.In the following simulations, the blue trajectories correspond to S 1 , while the red ones correspond to S 2 .
For a = −1.200,−1.300, −1.325, the 2D views of coexisting periodic attractors are shown in Figure 13.A typical chaotic attractor can be found at a = −1.4,which is shown in Figure 15.To illustrate the folding and stretching structure of the attractor, cross-sections (two-side Poincaré sections) are plotted in Figure 16.Further analysis shows that a boundary crisis occurs at a = −1.433.For the latter chaotic attractor, the Lyapunov exponents are 0.1042, 0, −0.3042, and thus the Kaplan-Yorke dimension is D KY = 2.3425.

Case D
Let us consider the system (2.9), which is denoted as Case D. It is obtained from system (2.6) by adding a small constant term 0.01 in the jerk function.In comparison to system (2.6), it is natural for system (2.9) to have some similar dynamical properties, such as the chaotic mechanism and multistability.Using the two initial conditions S 1 and S 2 , the bifurcation diagram of system (2.9) is shown in Figure 18.The bifurcation diagram of system (2.9) is also shown in Figure 19, in which the four initial conditions S 3 -S 6 are used.In Figure 19, there is a quite narrow range near a = −1.33 corresponding to two coexisting period four attractors and one chaotic attractor.For a = −1.29, the phase portrait of two coexisting asymmetric periodic attractors is shown in Figure 20.For a = −1.35, the phase portrait of two coexisting asymmetric chaotic attractors is shown in Figure 21.

Local bifurcation analysis
It is very known that both pitchfork bifurcation and Hopf bifurcation occur in the Lorenz system.These bifurcations and homoclinic bifurcation play important roles in locating chaotic dynamics.It is very interesting to see that both pitchfork bifurcation and Hopf bifurcation are also available in the jerk system (2.3), including both subcritical and supercritical versions.These two bifurcations can help us understand the mechanisms of chaotic attractors presented in Sections 3 and 4.

Pitchfork bifurcation at the origin for a = d
The following lemma is due to Sotomayor [57,58] and is the standard tool for proving the existence of a pitchfork bifurcation.Consider the following system: where g is sufficiently smooth, and µ is a parameter.As usual, Dg(x, µ) denotes the Jacobian matrix of the function g with respect to the variable x, and g µ (x, µ) represents the partial derivative of g with respect to µ.
Lemma 5.1.Consider the system (5.1).When µ = µ 0 , assume that there is an equilibrium x 0 , for which the following hypotheses are satisfied: (i) The Jacobian matrix Dg(x 0 , µ 0 ) has a simple eigenvalue 0 with right eigenvector v and left eigenvector w.It also has k eigenvalues with negative real part and n − k − 1 eigenvalues with positive real part.
Then, the system experiences a pitchfork bifurcation at the equilibrium x 0 as the parameter µ passes through µ = µ 0 .
Let us return to system (2.3), which always has a trivial equilibrium located at the origin.For it also has a pair of symmetric equilibria P 1,2 = (±x 0 , 0, 0), where sinh(x 0 ) For a d ≤ 1, there is no other equilibrium except the origin.
The Jacobian matrix of the system at the origin is whose characteristic equation is Theorem 5.1.Assume that the conditions (2.4) hold.If a ∈ (d, d + cb 2 ), then the origin is an asymptotically stable equilibrium of system (2.3).
Proof.According to the Routh-Hurwitz criterion and the assumption (2.4), all the eigenvalues of J(a) have negative real parts if and only if d < a < d + cb 2 .Thus, the conclusion follows.□ Theorem 5.2.For system (2.3), a pitchfork bifurcation occurs at the origin as a passes through a = d.Moreover, if d < 0, the bifurcation is supercritical, while if d > 0, it is subcritical.
Proof.Take a as the bifurcation parameter.Let f (x, a) be the vector field of system (2.3) with x = (x, y, z).
When a = d, the origin is the unique equilibrium of the system.In this case, the Jacobian matrix of f at this point is A direct computation gives According to Lemma 5.1, a pitchfork bifurcation occurs at the origin for a = d.Furthermore, according to [57], this bifurcation is supercritical if d < 0, while it is subcritical if d > 0. □

Hopf bifurcation at the origin for a
It is well known that Hopf bifurcation is a typical phenomenon related to the appearance or disappearance of limit cycles.It can be studied through some characteristic quantities, such as bifurcation formulas [59], Lyapunov coefficients [60] and focus quantities [61,62].For system (2.3), in order to obtain the direction of bifurcation and the period, amplitude, stability of limit cycles, we use the method of Hassard et al. [59] to get the results.
Before we state the results, let us introduce a notation: (5.5) Theorem 5.3.Assume that the conditions (2.4) hold.Then, for system (2.3), the following statements hold: (i) A Hopf bifurcation occurs at the origin as a passes through a = a 0 .
(ii) If d > 0, the Hopf bifurcation is supercritical, leading to a family of stable limit cycles for a > a 0 and a − a 0 being sufficiently small.
(iii) If d < 0, the Hopf bifurcation is subcritical, leading to a family of unstable limit cycles for a < a 0 and a 0 − a being sufficiently small.
(iv) The period of the limit cycle can be expanded in the form (v) The limit cycles are approximated by where A = 8(a − a 0 ) d .
Proof.In view of (5.2) and (5.3), according to the Proposition in [63], a Hopf bifurcation may occur at the origin when a passes through a = a 0 .When a = a 0 , the eigenvalues of the Jacobian matrix at the origin are λ 1,2 = ±bi, λ 3 = −c.In order to prove (i), it suffices to verify the transversality condition.From (5.3), we have By the assumption (2.4), we have indicating that the transversality condition is satisfied.Hence, the statement (i) holds.
The other statements are based on the computation of µ 2 , τ 2 and β 2 [59].Setting a = a 0 and using the transformation Now, we are in the position for applying the algorithm of Hassard et al. [59].Using the same notation as in [59], by computation we have Therefore, we have .
Based on these three quantities and Hopf bifurcation theory [59], the conclusions (ii)-(v) hold.□ 6. PSIM simulations PSIM (power simulation) is one of the fastest simulators for power electronics simulations.We use PSIM circuit simulation to validate the chaotic behavior of the following system: for which the numerical chaotic attractor was shown in Figure 5.
The circuit model of the system is shown in Figure 22.It has three operational channels, three integrators, two inverters and one hyperbolic sine converter.More specifically, the model has ten resistors with the following values: Let X, Y, Z be the output voltages of capacitors C 1 , C 2 and C 3 , respectively.By applying Kirchhoff's circuit laws to the circuit model, the state equations for X, Y, Z are described by The initial values used for the simulations are X(0) = Y(0) = 0 V, Z(0) = 0.1 V. Using PSIM 9.0.3, the phase portraits of the chaotic attractor are simulated and are shown in Figure 23.The results show that the circuit simulations are consistent with the results of numerical ones in Figure 5.

Conclusions
Jerk systems are very attractive due to simple forms and rich dynamical behaviors.More importantly, such systems can be realized with compact electrical circuit structures.In this paper, a jerk system with a hyperbolic sine nonlinearity and five parameters is proposed and explored.For the case with ε = 0, the jerk system displays very rich local bifurcation structures, which are rigorously treated in Section 5.Such local bifurcations are important for the analysis of the transitions from simple to complex dynamics in the general jerk systems.The novelty of this paper lies in the discovery of mechanisms leading to chaotic attractors in four jerk systems.Simultaneously, the corresponding multistability problems are also discussed.Two mechanisms with different bifurcation processes are investigated in detail for the four systems.In the first mechanism, the continuation of a pitchfork bifurcation (which is followed by a Hopf bifurcation) gives rise to forward period doubling cascades to chaos.In the second mechanism, the continuation of a pitchfork bifurcation (which is followed by a Hopf bifurcation) gives rise to backward period doubling cascades to chaos.In fact, the numerical findings in this paper rely on some very basic local bifurcations of the origin, which are rigorously treated in Section 5.In general, local bifurcations are of great importance for understanding the complex phenomena of chaotic mechanisms and multistabilities.Using PSIM, the circuit implementation is carried out, which verifies the chaotic response of a jerk system through simulations.For future studies, we hope the jerk circuit model in this paper can be used in secure communications, electronic circuits, random number generators and image encryption.
(a) x − y plane (b) y − z plane (c) x − z plane

Figure 15 .
Figure 15.2D views of the chaotic attractor of system (2.6) for a = −1.4.