A novel autonomous 5-D hyperjerk RC circuit with hyperbolic sine function is proposed in this paper. Compared to some existing 5-D systems like the 5-D Sprott B system, the 5-D Lorentz, and the Lorentz-like systems, the new system is the simplest 5-D system with complex dynamics reported to date. Its simplicity mainly relies on its nonlinear part which is synthetized using only two semiconductor diodes. The system displays only one equilibrium point and can exhibit both periodic and chaotic dynamical behavior. The complex dynamics of the system is investigated by means of bifurcation analysis. In particular, the striking phenomenon of multistability is revealed showing up to seven coexisting attractors in phase space depending solely on the system’s initial state. To the best of author’s knowledge, this rich dynamics has not yet been revealed in any 5-D dynamical system in general or particularly in any hyperjerk system. Pspice circuit simulations are performed to verify theoretical/numerical analysis.
The study of three-dimensional dynamical systems seems to be mature [
This paper investigates the dynamics of a novel 5-D hyperjerk circuit with a very simple nonlinear part (a pair of semiconductor diodes). The new circuit can be regarded as a 5-D version of the jerk circuit previously reported by Kengne and collaborators [
The paper is organized as follows: Section
The electronic circuit of the oscillator under investigation is depicted in Figure
Electronic circuit of the proposed 5-D hyperjerk system.
System (
To produce phase portraits, bifurcation diagrams, and Lyapunov spectrum, the system was solved using the classical fourth-order Runge-Kutta algorithm with the time step always
To reveal the type of transition leading to chaos, a single control parameter (b) was considered to vary in the range
Backward continuation of system (
Numerical phase space trajectories (left) and Pspice based simulation results (right) showing the classical period doubling routes to chaos in the novel 5-D system.
As the system experiences the classical period doubling route to chaos, it is obvious that antimonotonicity can be observed. Represented on Figure
Bifurcation diagrams showing local maxima of the coordinate
Multistability has been previously revealed in many dynamical systems [
With reference to the bifurcation diagram of Figure
Bifurcation diagram for illustrating the coexistence of disconnected chaotic attractors with a pair of period-2 limit cycle. The diagram is plotted by forward or backward continuation of parameter b with the following initial conditions x1(0) = 3; x2(0) = x3(0) = x4(0) = x5(0) = 0 and x1(0) = 1; x2(0) = x3(0) = x4(0) = x5(0) = 0.
Two-dimensional projections (x4-x5) of four coexisting chaotic and periodic attractors for b = 29.57, a0 = a2 = a3, a1 = 7, a4 = 5.4433 x 10−4. Initial conditions are indicated in Table
Comparative analysis of some dynamical systems by using the largest Lyapunov exponent (
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Initial conditions for the abundant coexisting attractors.
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| Four disconnected chaotic attractors | | | | – |
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| Four disconnected chaotic and periodic attractors | | | | |
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| Five disconnected chaotic and periodic attractors | | | | |
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| Five disconnected periodic attractors | | | | |
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| Six disconnected chaotic and periodic attractors | | | | |
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| Seven disconnected chaotic and periodic attractors | | | | – |
To observe more than four different attractors in the system, the second method described above (see Section
Bifurcation diagram for justifying the coexistence of four, five, and six different attractors in the phase space. The diagram in blue and red is plotted by forward and backward continuation while the diagram in green is plotted by following the attractor defined at a2 = 2.71 for x1(0) = 3; x2(0) = x3(0) = x4(0) = x5(0) = 0.
Coexistence of four different attractors (a pair of period-1 limit cycles and two different symmetric attractors) for a2 = 2.458 with the rest of system’s parameters as follows: b = a1 = 3, a0 = 1.75, a3 = 1, a4 = 0.0054. Initial states are given in Table
If systems parameters are
Coexistence of five different period-1 limit cycles for a2 = 3 with the rest of system’s parameters as follows: b = a1 = 3, a0 = 1.75, a3 =1, a4 = 0.0054. Initial states are given in Table
For
Two-dimensional projections (x1-x2) of five coexisting attractors for a2 = 2.71 (a pair of chaotic attractors, a pair of period-1 limit cycle, and a symmetric chaotic attractor) with the rest of system’s parameters as follows: b = a1 = 3, a0 = 1.75, a3 = 1, a4 = 0.0054. Initial states are given in Table
For
Two-dimensional projections (x1-x2) of six coexisting attractors for a2 = 2.8 (two pairs of chaotic attractors and a pair of period-1 limit cycle) with the rest of system’s parameters as follows: b = a1 = 3, a0 = 1.75, a3 = 1, a4 = 0.0054. Initial states are given in Table
Bifurcation diagram for illustrating the coexistence of seven different attractors in the phase space. The diagrams are plotted using the same methods as in Figure
More interestingly, another window of multiple coexisting attractors can be revealed when
Coexistence of seven disconnected attractors (two pairs of chaotic attractors, a pair of period-1 limit cycle, and a symmetric period-1 limit cycle) for a2 = 3 with the rest of system’s parameters as follows: b = a1 = 3, a0 = 1.75, a3 = 1, a4 = 0.0109. Initial states are given in Table
This work represents an enriching contribution to the understanding of the nonlinear dynamics of this type of oscillators [
Our motivation in this section is to verify the theoretical/numerical results obtained previously by performing some Pspice based simulations of the circuit. Furthermore, it is important to evaluate the effects of simplifying assumptions (e.g., ideal diode model and ideal op. amplifiers) considered during the mathematical modeling process, on the behavior of a hardware prototype of the 5-D hyperjerk circuit in Pspice. Briefly recall that an interesting aspect of using Pspice is the possibility of setting initial capacitors’ voltages and analyzing the corresponding influence on the dynamics of the complete electronic circuit. Thus, the presence of multiple coexisting attractors can be tracked in a straightforward manner.
First to report the reverse period doubling routes to chaos observed during the numerical analysis, the circuit of Figure
Secondly, coexistence of multiple attractors can also be confirmed by Pspice based simulations with the following electronic circuit components:
Pspice simulation results showing the coexistence of four different attractors for
Pspice simulation results showing the coexistence of five different attractors for
Pspice simulation results showing the coexistence of six different attractors for
A novel 5-D hyperjerk circuit with a very simple nonlinear part has been introduced in this work. The circuit is obtained by introducing additional feedback loops in the realization circuit of a jerk system previously reported by J. Kengne and collaborators. The modification yields the simplest 5-D hyperjerk system reported up to date. More interestingly, for some given sets of parameters, the system experiences a plethora of multiple coexisting attractors. For instance, up to seven disconnected attractors coexist in the system depending solely on the initial conditions. To the best of author’s knowledge, such dynamics has not yet been reported in any hyperjerk system and thus deserves dissemination. Pspice based simulations were carried out to support the theoretical analysis. A detailed exploration of the parameter space (both experimentally and numerically) in view of revealing hyperchaotic behavior and hidden attractors in system (
The authors declare that there are no conflicts of interest regarding the publication of this paper.