Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems
<p>Attractor of system (5) when <math display="inline"> <semantics> <mrow> <mi>q</mi> <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> <msub> <mi>q</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>. (<b>a</b>) <span class="html-italic">Y-R-P</span> plane; (<b>b</b>) <span class="html-italic">Y-R-M</span> plane; (<b>c</b>) Projection on <span class="html-italic">P-M</span> plane; (<b>d</b>) Projection on <span class="html-italic">R-P</span> plane.</p> "> Figure 2
<p>Phrase diagram for system (5) with commensurate order case.</p> "> Figure 3
<p>The operation state of each variable when <math display="inline"> <semantics> <mrow> <mi>q</mi> <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> <msub> <mi>q</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0.15</mn> </mrow> </semantics> </math>.</p> "> Figure 4
<p>Power spectrum of system (5) when <math display="inline"> <semantics> <mrow> <mi>q</mi> <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> <msub> <mi>q</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.15</mn> </mrow> </semantics> </math>.</p> "> Figure 5
<p>Phrase diagram for system (5) with <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0.999</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0.995</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>c</b>) Projection when <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0.995</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> on <span class="html-italic">P-M</span> plane; (<b>d</b>) Projection when <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0.995</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> on <span class="html-italic">R-P</span> plane.</p> "> Figure 6
<p>Phrase diagrams for system (5) with <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0.999</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0.97</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0.95</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0.905</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>e</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.95</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>f</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>g</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.995</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>h</b>) <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.85</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>.</p> "> Figure 7
<p>Phrase diagram and variable changes when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>2.5</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics> </math>. (<b>a</b>) Phrase diagram when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>; (<b>b</b>) Phrase diagram when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>; (<b>c</b>) Phrase diagram when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math>; (<b>d</b>) National income <math display="inline"> <semantics> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> changes when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>; (<b>e</b>) National income <math display="inline"> <semantics> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> changes when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math>; (<b>f)</b> National income <math display="inline"> <semantics> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> changes when <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>.</p> ">
Abstract
:1. Introduction
2. Construction of the Fractional Order IS–LM Model
2.1. Caputo Fractional Order Derivative Definition and Stability Judgment
The Stability of Fractional Order System
2.2. Economics Meaning of Caputo Fractional Derivative
2.3. Four-Dimensional IS–LM Macroeconomics Model
3. Theoretical Analysis of Fractional Order IS–LM System
3.1. Equilibrium Solution and Its Stability of the System
3.2. Dissipativity and the Existence of Attractor
4. Analysis on Dynamic Evolution of Fractional Order IS–LM System
4.1. Commensurate Order Case
Convergent State of the System
4.2. Incommensurate Order Case
- (1)
- , and changes, the state changes of system (5) are shown in Figure 5.
- (2)
- When change separately, the state change of system (5) is shown in Figure 6. Similar to the change state of , when , and reduces from 1 to 0, from Figure 6a–d, the running state of system (5) changes gradually. When , the system began to converge from the attractor, that is, the system convergence just entered the fractional order state, the four-dimensional macroeconomics system running stably, and under controlled conditions.
4.3. Chaos Control of IS–LM Fractional-Order System
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Ma, J.; Ren, W.; Zhan, X. Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems. Entropy 2016, 18, 332. https://doi.org/10.3390/e18090332
Ma J, Ren W, Zhan X. Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems. Entropy. 2016; 18(9):332. https://doi.org/10.3390/e18090332
Chicago/Turabian StyleMa, Junhai, Wenbo Ren, and Xueli Zhan. 2016. "Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems" Entropy 18, no. 9: 332. https://doi.org/10.3390/e18090332