Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems
<p>System (<a href="#FD1-entropy-23-00261" class="html-disp-formula">1</a>) with its chaotic hidden attractors exhibited in distinct planes according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p> "> Figure 2
<p>(<b>a</b>) The bifurcation diagram; and (<b>b</b>) LEs of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with commensurate order by varying <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>∈</mo> <mo>(</mo> <mn>0.90</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and the system’s parameters <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Chaotic hidden attractor of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with commensurate order <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.9747</mn> </mrow> </semantics></math>, shown on different planes according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and the system’s parameters <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Basin of attraction section <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> of attractors shown in <a href="#entropy-23-00261-f003" class="html-fig">Figure 3</a> for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> according to the initial condition of the third state variable <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Bifurcation diagrams of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with commensurate <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> through fixing <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and varying the parameters (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>(</mo> <mn>2.3</mn> <mo>,</mo> <mspace width="4pt"/> <mn>3.2</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mspace width="4pt"/> <mn>3.2</mn> <mo>)</mo> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <mo>(</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 6
<p>The phase portraits of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with its commensurate order <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> according to IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and the parameters’ values <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>±</mo> <mn>0.8</mn> </mrow> </semantics></math> on different projections (black plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, green plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.8</mn> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math>-plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </semantics></math>-plane; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-plane.</p> "> Figure 7
<p>The time series of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with its commensurate order <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math>, corresponding to its IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> and its parameters <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>±</mo> <mn>0.8</mn> </mrow> </semantics></math> (black plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, green plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.8</mn> </mrow> </semantics></math>): (<b>a</b>) the state-space variable <span class="html-italic">x</span>; (<b>b</b>) the state-space variable <span class="html-italic">y</span>; and (<b>c</b>) the state-space variable <span class="html-italic">z</span>.</p> "> Figure 8
<p>The chaotic bursting oscillation according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.985</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>3.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>: (<b>a</b>) the time series of the state-space variable <span class="html-italic">x</span>; (<b>b</b>) the phase portrait in <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </semantics></math>-plane; and (<b>c</b>) the phase portrait in <math display="inline"><semantics> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-plane.</p> "> Figure 9
<p>Passing transition behavior according to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.985</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>3.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>: (<b>a</b>) the corresponding time series of the state-space variable <span class="html-italic">x</span>; and (<b>b</b>) the phase portrait in 3D projection.</p> "> Figure 10
<p>(<b>a</b>) The diagram of bifurcation of the FoS given in (5) for <span class="html-italic">q</span> ∈ (0.9, 1) with two set of ICs, (0, 0, 0) (blue plot) and (0.5, 1, −0.2) (red plot); (<b>b</b>) two coexisting hidden attractors for <span class="html-italic">q</span> = 0.98 (arrow L in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>a) and <span class="html-italic">q</span> = 0.9882 (arrow R in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>a) corresponding to two set of ICs, (0, 0, 0) (blue plot) and (0.5, 1, −0.2) (red plot); and (<b>c</b>) basin of attraction section x − y of attractors shown in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>b (arrow L), for <span class="html-italic">q</span> = 0.98, and an initial condition in the third state variable z = 0. The colors shown in the figure associate with the colors of the attractors given in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>b (arrow L).</p> "> Figure 10 Cont.
<p>(<b>a</b>) The diagram of bifurcation of the FoS given in (5) for <span class="html-italic">q</span> ∈ (0.9, 1) with two set of ICs, (0, 0, 0) (blue plot) and (0.5, 1, −0.2) (red plot); (<b>b</b>) two coexisting hidden attractors for <span class="html-italic">q</span> = 0.98 (arrow L in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>a) and <span class="html-italic">q</span> = 0.9882 (arrow R in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>a) corresponding to two set of ICs, (0, 0, 0) (blue plot) and (0.5, 1, −0.2) (red plot); and (<b>c</b>) basin of attraction section x − y of attractors shown in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>b (arrow L), for <span class="html-italic">q</span> = 0.98, and an initial condition in the third state variable z = 0. The colors shown in the figure associate with the colors of the attractors given in <a href="#entropy-23-00261-f010" class="html-fig">Figure 10</a>b (arrow L).</p> "> Figure 11
<p>(<b>a</b>) Multiple coexisting hidden attractors for three ICs, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>0.2</mn> <mo>,</mo> <mspace width="4pt"/> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math>; and (<b>b</b>) basin of attraction section <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> of attractors shown in <a href="#entropy-23-00261-f011" class="html-fig">Figure 11</a>a for <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> according to the IC of the third state variable <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The colors of the figure associate with the colors of the attractors given in <a href="#entropy-23-00261-f011" class="html-fig">Figure 11</a>a.</p> "> Figure 12
<p>(<b>a</b>) The diagram of bifurcation; and (<b>b</b>) the LEs of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with incommensurate order by varying <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0.80</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and fixing <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 13
<p>(<b>a</b>) The diagram of bifurcation; and (<b>b</b>) the LEs of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with incommensurate order by varying <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and fixing <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 14
<p>(<b>a</b>) The diagram of bifurcation; and (<b>b</b>) the LEs of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with incommensurate order by varying <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0.75</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and fixing <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 15
<p>The diagrams of LEs of incommensurate system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) as function of <math display="inline"><semantics> <mi>γ</mi> </semantics></math> for: (<b>a</b>) commensurate order <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math>; (<b>b</b>) incommensurate order <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) incommensurate order <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>; and (<b>d</b>) incommensurate order <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.99</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 16
<p>The phase portraits of system (<a href="#FD5-entropy-23-00261" class="html-disp-formula">5</a>) with the incommensurate orders <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, in accordance with the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> and the parameter <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>±</mo> <mn>0.8</mn> </mrow> </semantics></math> on distinct projections (black plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, green plot for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.8</mn> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math>-plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </semantics></math>-plane; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-plane.</p> "> Figure 17
<p>Passing transition behavior by taking the orders <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>; the parameters <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>3.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>; and the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>: (<b>a</b>) the corresponding time series of the state-space variable <span class="html-italic">x</span>; and (<b>b</b>) the phase portrait in 3D projection.</p> "> Figure 18
<p>Multiple coexisting hidden attractors according to the three ICs <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mo>−</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 19
<p>Propagating of the variable one-scroll chaotic hidden attractor on a line corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math>: (<b>a</b>) <span class="html-italic">x</span>-line when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mo>±</mo> <mn>5</mn> </mrow> </semantics></math>; (<b>b</b>) <span class="html-italic">y</span>-line when <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mo>±</mo> <mn>2</mn> </mrow> </semantics></math>; and (<b>c</b>) <span class="html-italic">z</span>-line when <math display="inline"><semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> </mrow> </semantics></math>.</p> "> Figure 20
<p>Propagating of the variable one-scroll chaotic hidden attractor on a line corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.98</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>: (<b>a</b>) <span class="html-italic">x</span>-line when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mo>±</mo> <mn>5</mn> </mrow> </semantics></math>; (<b>b</b>) <span class="html-italic">y</span>-line when <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mo>±</mo> <mn>2</mn> </mrow> </semantics></math>; and (<b>c</b>) <span class="html-italic">z</span>-line when <math display="inline"><semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ℓ</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> </mrow> </semantics></math>.</p> "> Figure 21
<p>Propagating of the variable one-scroll chaotic hidden attractor on a lattice corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ω</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 22
<p>Propagating of the variable one-scroll chaotic hidden attractor on a lattice corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.98</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math>-lattice when <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ω</mi> <mo>)</mo> <mo>=</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 23
<p>Propagating of the variable one-scroll chaotic hidden attractor on a 3D <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-grid corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ω</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 24
<p>Propagating of the variable one-scroll chaotic hidden attractor on a 3D <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math>-grid corresponding to the IC <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, and according to <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2.8</mn> </mrow> </semantics></math>,and <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.98</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ω</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 25
<p>Basin of attraction section <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> of many attractors shown in <a href="#entropy-23-00261-f024" class="html-fig">Figure 24</a> for <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mn>0.98</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> according to the IC of the third state variable <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The colors in this figure associate with the colors of the attractors given in <a href="#entropy-23-00261-f024" class="html-fig">Figure 24</a>.</p> ">
Abstract
:1. Introduction
2. A Non-Equilibrium FoS
3. The Commensurate FoS
3.1. Chaos vs. the Variety in the Fractional-Order Values
3.2. Chaos vs. the Variety in the Values of System’s Parameters
3.3. Inversion Property
3.4. Hidden Bursting Oscillation
3.5. Coexisting Hidden Attractors
4. Incommensurate FoS
5. Variable-Boostable Hidden Attractors of Commensurate and Incommensurate FoS
5.1. State 1: A Line of Variable Hidden Attractors
- *
- *
- *
5.2. State 2: A Lattice of Variable Hidden Attractors
- *
- *
- *
5.3. State 3: A 3D Grid of Variable Hidden Attractors
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Chen, L.; Pan, W.; Wang, K.; Wu, R.; Tenreiro Machado, J.A.; Lopes, A.M. Generation of a family of fractional order hyper-chaotic multi-scroll attractors. Chaos Solitons Fractals 2017, 105, 244–255. [Google Scholar] [CrossRef]
- Grigorenko, I.; Grigorenko, E. Chaotic dynamics of the fractional lorenz system. Phys. Rev. Lett. 2003, 91, 034101. [Google Scholar] [CrossRef]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
- Cafagna, D. Past and present-Fractional calculus: A mathematical tool from the past for present engineers. IEEE Ind. Electron. Mag. 2007, 1, 35–40. [Google Scholar] [CrossRef]
- Zhao, J.; Wang, S.; Chang, Y.; Li, X. A novel image encryption scheme based on an improper fractional-order chaotic system. Nonlinear Dyn. 2015, 80, 1721–1729. [Google Scholar] [CrossRef]
- Hu, F.; Chen, L.C.; Zhu, W.Q. Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation. Int. J. Non-Linear Mech. 2012, 47, 1081–1087. [Google Scholar] [CrossRef]
- Sundarapandian, V.; Pehlivan, I. Analysis, control, synchronization, and circuit design of a novel chaotic system. Math. Comput. Model. 2012, 55, 1904–1915. [Google Scholar] [CrossRef]
- Zhang, S.; Zeng, Y.; Li, Z.; Zhou, C. Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium. Int. J. Bifurc. Chaos 2018, 28, 1850167. [Google Scholar] [CrossRef]
- Sun, K.; Sprott, J.C. Bifurcations of fractional-order diffusionless lorenz system. Electron. J. Theor. Phys. 2009, 6, 123–134. [Google Scholar]
- Hartley, T.T. Chaos in a fractional order Chua. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 1995, 42, 485–490. [Google Scholar] [CrossRef]
- Hajipoor, A.; Shandiz, H.T.; Marvi, H. Dynamic analysis of the fractional-order chen chaotic system. World Appl. Sci. J. 2009, 7, 109–115. [Google Scholar]
- Zhou, P.; Huang, K. A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 2005–2011. [Google Scholar] [CrossRef]
- Pham, V.T.; Ouannas, A.; Volos, C.; Kapitaniak, T. A simple fractional-order chaotic system without equilibrium and its synchronization. AEU-Int. J. Electron. Commun. 2018, 86, 69–76. [Google Scholar] [CrossRef]
- Cafagna, D.; Grassi, G. Elegant chaos in fractional-order system without equilibria. Math. Probl. Eng. 2013, 2013, 1–7. [Google Scholar] [CrossRef]
- Leonov, G.A.; Kuznetsov, N.V.; Vagaitsev, V.I. Localization of hidden Chua’s attractors. Phys. Lett. A 2011, 375, 2230–2233. [Google Scholar] [CrossRef]
- Shahzad, M.; Pham, V.T.; Ahmad, M.A.; Jafari, S.; Hadaeghi, F. Synchronization and circuit design of a chaotic system with coexisting hidden attractors. Eur. Phys. J. Spec. Top. 2015, 224, 1637–1652. [Google Scholar] [CrossRef]
- Chaudhuri, U.; Prasad, A. Complicated basins and the phenomenon of amplitude death in coupled hidden attractors. Phys. Lett. A 2014, 378, 713–718. [Google Scholar] [CrossRef]
- Tchinda, S.T.; Mpame, G.; Takougang, A.N.; Tamba, V.K. Dynamic analysis of a snap oscillator based on a unique diode nonlinearity effect, offset boosting control and sliding mode control design for global chaos synchronization. J. Control Autom. Electric. Syst. 2019, 30, 970–984. [Google Scholar] [CrossRef]
- Wang, X.; Pham, V.T.; Jafari, S.; Volos, C.; Munoz-Pacheco, J.M.; Tlelo-Cuautle, E. A new chaotic system with stable equilibrium: From theoretical model to circuit implementation. Phys. Lett. A 2017, 5, 8851–8858. [Google Scholar] [CrossRef]
- Li, C.B.; Sprott, J.C. Variable-boostable chaotic flows. Optik 2016, 127, 10389–10398. [Google Scholar] [CrossRef]
- Bayani, A.; Rajagopal, K.; Khalaf, A.J.M.; Jafari, S.; Leutcho, G.D.; Kengne, J. Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity, coexisting multiple attractors, and offset boosting. Phys. Lett. A 2019, 383, 1450–1456. [Google Scholar] [CrossRef]
- Pham, V.T.; Wang, X.; Jafari, S.; Volos, C.; Kapitaniak, T. From Wang–Chen system with only one stable equilibrium to a new chaotic system without equilibrium. Int. J. Bifurcat. Chaos 2017, 27, 1750097. [Google Scholar] [CrossRef]
- Li, C.; Sprott, J.C.; Mei, Y. An infinite 2-D lattice of strange attractors. Nonlinear Dyn. 2017, 89, 2629–2639. [Google Scholar] [CrossRef]
- Munoz-Pacheco, J.M.; Zambrano-Serrano, E.; Volos, C.; Tacha, O.I.; Stouboulos, N.; Pham, V.T. A fractional order chaotic system with a 3D grid of variable attractors. Chaos Solitons Fractals 2018, 113, 69–78. [Google Scholar] [CrossRef]
- Zhang, S.; Wang, X.; Zeng, Z. A simple no-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation. Chaos 2020, 30, 053129. [Google Scholar] [CrossRef] [PubMed]
- Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
- Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002, 29, 3–22. [Google Scholar] [CrossRef]
- Diethelm, K. The Analysis of Fractional Differential Equations, an Application Oriented Exposition Using Differential Operators of Caputo Type; Springer: Berlin, Germany, 2010. [Google Scholar]
- Danca, M.F.; Kuznetsov, N. Matlab Code for Lyapunov Exponents of Fractional-Order Systems. Int. J. Bifurc. Chaos 2018, 28, 1850067. [Google Scholar] [CrossRef] [Green Version]
- Sun, K.; Sprott, J.C. Periodically Forced Chaotic System With Signum Nonlinearity. Int. J. Bifurc. Chaos 2010, 20, 1499–1507. [Google Scholar] [CrossRef] [Green Version]
- Gans, R.F. When is cutting chaotic. J. Sound Vibr. 1995, 188, 75–83. [Google Scholar] [CrossRef]
- Bao, B.C.; Wu, P.; Bao, H.; Chen, M.; Xu, Q. Chaotic bursting in memristive diode bridge-coupled Sallen-Key lowpass filter. Electron. Lett. 2017, 53, 1104–1105. [Google Scholar] [CrossRef]
- Han, X.; Yu, Y.; Zhang, C.; Xia, F.; Bi, Q. Turnover of hysteresis determines novel bursting in duffing system with multiple-frequency external forcings. Int. J. Non-Linear Mech. 2017, 89, 69–74. [Google Scholar] [CrossRef]
- Wang, M.J.; Liao, X.H.; Deng, Y.; Li, Z.J.; Zeng, Y.C.; Ma, M.L. Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System With Coexisting Hidden Attractors. J. Comput. Nonlinear Dynam. 2019, 14, 071002. [Google Scholar] [CrossRef]
- Kingni, S.T.; Nana, B.; Ngueuteu, G.M.; Woafo, P.; Danckaert, J. Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect. Chaos Solitons Fractals 2015, 71, 29–40. [Google Scholar] [CrossRef]
- Echenausía-Monroy, J.L.; Gilardi-Velázquez, H.E.; Jaimes-Reátegui, R.; Aboites, V.; Huerta-Cuellar, G. A physical interpretation of fractional-order-derivatives in a jerk system: Electronic approach. Commun. Nonlinear Sci. Numer. Simul. 2020, 90, 105413. [Google Scholar] [CrossRef]
- Echenausía-Monroy, J.L.; Huerta-Cuellar, G.; Jaimes-Reátegui, R.; García-López, J.H.; Aboites, V.; Cassal-Quiroga, B.B.; Gilardi-Velázquez, H.E. Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System. Electronics 2020, 9, 880. [Google Scholar] [CrossRef]
q | Dynamic State |
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Period 1 | |
Period 2 | |
Period 4 | |
quasiperiodic | |
chaos | |
periodic-route | |
chaos |
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Debbouche, N.; Momani, S.; Ouannas, A.; Shatnawi, ’.T.; Grassi, G.; Dibi, Z.; Batiha, I.M. Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems. Entropy 2021, 23, 261. https://doi.org/10.3390/e23030261
Debbouche N, Momani S, Ouannas A, Shatnawi ’T, Grassi G, Dibi Z, Batiha IM. Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems. Entropy. 2021; 23(3):261. https://doi.org/10.3390/e23030261
Chicago/Turabian StyleDebbouche, Nadjette, Shaher Momani, Adel Ouannas, ’Mohd Taib’ Shatnawi, Giuseppe Grassi, Zohir Dibi, and Iqbal M. Batiha. 2021. "Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems" Entropy 23, no. 3: 261. https://doi.org/10.3390/e23030261