Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode
This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to fo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/11/1803 |
| _version_ | 1797549264246669312 |
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| author | Pattrawut Chansangiam |
| author_facet | Pattrawut Chansangiam |
| author_sort | Pattrawut Chansangiam |
| collection | DOAJ |
| description | This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> diverges in a spiral form but <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> converges to the equilibrium point for any initial point <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor. |
| first_indexed | 2024-03-10T15:12:11Z |
| format | Article |
| id | doaj.art-a058796f067243ecab7426dfd1c4f7a3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T15:12:11Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-a058796f067243ecab7426dfd1c4f7a32023-11-20T19:15:15ZengMDPI AGSymmetry2073-89942020-10-011211180310.3390/sym12111803Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s DiodePattrawut Chansangiam0Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, ThailandThis paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> diverges in a spiral form but <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> converges to the equilibrium point for any initial point <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>. Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.https://www.mdpi.com/2073-8994/12/11/1803chaos theoryelectrical circuit analysisjerk circuitChua’s diodesystem of differential equationshidden attractor |
| spellingShingle | Pattrawut Chansangiam Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode Symmetry chaos theory electrical circuit analysis jerk circuit Chua’s diode system of differential equations hidden attractor |
| title | Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode |
| title_full | Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode |
| title_fullStr | Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode |
| title_full_unstemmed | Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode |
| title_short | Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode |
| title_sort | three saddle foci chaotic behavior of a modified jerk circuit with chua s diode |
| topic | chaos theory electrical circuit analysis jerk circuit Chua’s diode system of differential equations hidden attractor |
| url | https://www.mdpi.com/2073-8994/12/11/1803 |
| work_keys_str_mv | AT pattrawutchansangiam threesaddlefocichaoticbehaviorofamodifiedjerkcircuitwithchuasdiode |