# On the n-Dimensional Phase Portraits

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## Abstract

**:**

## Featured Application

**The current graphic analysis can be applied during the mathematical analysis of high order systems, for instance, power electronic, mechanical, aeronautic, and nuclear plant systems among others.**

## Abstract

## 1. Introduction

## 2. The n-Dimensional Phase Portraits by State Combinations

#### 2.1. Foundation

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

#### 2.2. Procedure

**Procedure**

**1.**

- 1.
- Construct the state space representation of the system.
- 2.
- Define regular initial conditions’ values for each state ().
- 3.
- Construct a phase portrait for each combination of states, a total of$\left(\frac{n!}{2(n-2)!}\right)$.
- 4.
- If necessary, redefine the initial conditions’ values and repeat step 3; that is, get enough sharpness.
- 5.
- Analyze each phase portrait separately.

#### 2.3. Examples

${x}_{1}\left(0\right)$ | ${x}_{2}\left(0\right)$ | ${x}_{3}\left(0\right)$ |

−30 | −30 | −30 |

−30 | −30 | −20 |

−30 | −20 | −30 |

−30 | −20 | −20 |

⋮ | ⋮ | ⋮ |

30 | 30 | 20 |

30 | 30 | 30 |

${x}_{1}\left(0\right)$ | ${x}_{2}\left(0\right)$ | ${x}_{3}\left(0\right)$ | ${\dot{x}}_{1}$ | ${\dot{x}}_{2}$ | ${\dot{x}}_{3}$ | $len({\dot{x}}_{1}$, ${\dot{x}}_{2})$ | $len({\dot{x}}_{1}$, ${\dot{x}}_{3})$ | $len({\dot{x}}_{2}$, ${\dot{x}}_{3})$ |

−30 | −30 | −30 | 30 | 30 | 30 | 42.4 | 42.4 | 42.4 |

−30 | −30 | −20 | 30 | 30 | 20 | 42.4 | 36 | 36 |

−30 | −20 | −30 | 30 | 20 | 30 | 36 | 42.4 | 42.4 |

−30 | −20 | −20 | 30 | 20 | 20 | 36 | 36 | 28.3 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

30 | 30 | 20 | −30 | −30 | −20 | 42.4 | 36 | 36 |

30 | 30 | 30 | −30 | −30 | −30 | 42.4 | 42.4 | 42.4 |

## 3. The n-Dimensional Phase Portraits by Coordinates’ Transformations

#### 3.1. Foundation

#### 3.2. Procedure

**Procedure**

**2.**

- 1.
- Construct the state space representation of the system.
- 2.
- Define regular initial conditions values for each state.
- 3
- Linearize if necessary, in every combination of regular initial conditions; alternatively linearize only in an equilibrium point of interest.
- 4.
- Try to found matrices such that$AT=T\Lambda $; if not possible use another method for order reduction.
- 5.
- Determine the dominant states and construct a phase portrait for each combination of them.
- 6.
- If necessary, redefine the initial conditions values and repeat from step 3; that is, get enough sharpness.
- 7.
- Analyze each phase portrait separately.

#### 3.3. Examples

## 4. The State-by-State n-Dimensional Phase Portraits

#### 4.1. Foundation

#### 4.2. Procedure

**Procedure**

**3.**

- 1.
- Construct the state space representation of the system.
- 2.
- Define regular initial conditions values for each state.
- 3.
- Calculate${f}_{i}\left(x\left(0\right)\right)$for each i-th state, and for the set of initial conditions.
- 4.
- Plot$[{x}_{i}\left(0\right),{x}_{i}\left(0\right){f}_{i}\left(x\left(0\right)\right)]$
- 5.
- Repeat Step 3 for each state, a total of n.
- 6.
- If necessary, redefine the initial conditions’ values and repeat from step 5; that is, get enough sharpness.
- 7.
- Analyze each phase portrait separately.

#### 4.3. Examples

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Matlab Code to Generate a n-Dimensional Phase Portrait by State Combinations

- %%%%%%%%%%%%%%%%%
`Quiver scale`%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `scale=15;`- %%%%%%%%%%%%%%%%%
`Generate grid`%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `x=permn(-3:3/15:3,4); %(c) Jos van der Geest library from Matlab`- %%%%%%%%%%%%%%%%%
`THIS IS THE DYNAMICS`%%%%%%%%%%%%%%%%%%%%%%%%% `f=[sin(x(:,1)),cos(x(:,2)),tan(x(:,3)/10),abs(x(:,4))];`- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
`close all``figure(’Name’,’Phase Portrait x_1 x_2’,’NumberTitle’,’off’)``hold on``quiver(x(:,1),x(:,2),f(:,1),f(:,2),scale,’k’)``xlabel(’x_1’,’FontSize’,18)``ylabel(’x_2’,’FontSize’,18)``grid on``figure(’Name’,’Phase Portrait x_1 x_3’,’NumberTitle’,’off’)``hold on``scale=30;``quiver(x(:,1),x(:,3),f(:,1),f(:,3),scale,’k’)``xlabel(’x_1’,’FontSize’,18)``ylabel(’x_3’,’FontSize’,18)``grid on``figure(’Name’,’Phase Portrait x_1 x_4’,’NumberTitle’,’off’)``hold on``scale=30;``quiver(x(:,1),x(:,4),f(:,1),f(:,4),scale,’k’)``xlabel(’x_1’,’FontSize’,18)``ylabel(’x_4’,’FontSize’,18)``grid on``figure(’Name’,’Phase Portrait x_2 x_3’,’NumberTitle’,’off’)``hold on``scale=30;``quiver(x(:,2),x(:,3),f(:,2),f(:,3),scale,’k’)``xlabel(’x_2’,’FontSize’,18)``ylabel(’x_3’,’FontSize’,18)``grid on``figure(’Name’,’Phase Portrait x_2 x_4’,’NumberTitle’,’off’)``hold on``scale=30;``quiver(x(:,2),x(:,4),f(:,2),f(:,4),scale,’k’)``xlabel(’x_2’,’FontSize’,18)``ylabel(’x_4’,’FontSize’,18)``grid on``figure(’Name’,’Phase Portrait x_3 x_4’,’NumberTitle’,’off’)``hold on``scale=30;``quiver(x(:,3),x(:,4),f(:,3),f(:,4),scale,’k’)``xlabel(’x_3’,’FontSize’,18)``ylabel(’x_4’,’FontSize’,18)``grid on``shg`

## Appendix B. Matlab Code to Generate a State by State n-Dimensional Phase Portrait

- %%%%%%%%%%%%%%%%%
`Quiver scale`%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `scale=1;`- %%%%%%%%%%%%%%%%%
`Generate grid`%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `x=permn(-3:3/10:3,4); %(c) Jos van der Geest library from Matlab`- %%%%%%%%%%%%%%%%%
`THIS IS THE DYNAMICS`%%%%%%%%%%%%%%%%%%%%%%%%% `f=[sin(x(:,1)),cos(x(:,2)),tan(x(:,3)/10),abs(x(:,4))];`- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
`s=size(x(:,1));``vy=zeros(s(:,1),1);``z=1:1:s(:,1);``z=z’;``close all``subplot(4,1,1)``quiver( x(:,1) , y , x(:,1) , x(:,1).∗f(:,1) ,scale,’k’,’ShowArrowHead’,``’off’,’MaxHeadSize’,0.05,’AlignVertexCenters’,’on’,’LineWidth’,1)``hold on;``line([-5 5],[0 0],’Color’,’k’)``subplot(4,1,2)``quiver( x(:,2) , y , x(:,2) , x(:,2).∗f(:,2) ,scale,’b’,’ShowArrowHead’,``’off’,’MaxHeadSize’,0.05,’AlignVertexCenters’,’on’,’LineWidth’,1)``hold on;``line([-5 5],[0 0],’Color’,’b’)``subplot(4,1,3)``quiver( x(:,3) , y , x(:,3) , x(:,3).∗f(:,3) ,scale,’r’,’ShowArrowHead’,``’off’,’MaxHeadSize’,0.05,’AlignVertexCenters’,’on’,’LineWidth’,1)``hold on;``line([-5 5],[0 0],’Color’,’r’)``subplot(4,1,4)``quiver( x(:,4) , y , x(:,4) , x(:,4).∗f(:,4) ,scale,’m’,’ShowArrowHead’,``’off’,’MaxHeadSize’,0.05,’AlignVertexCenters’,’on’,’LineWidth’,1)``hold on;``line([-5 5],[0 0],’Color’,’m’)``figure``vsubplot(4,1,1)``vplot(x(:,1),x(:,1).∗f(:,1),’k’)``subplot(4,1,2)``plot(x(:,2),x(:,2).∗f(:,2),’b’)``subplot(4,1,3)``plot(x(:,3),x(:,3).∗f(:,3),’r’)``subplot(4,1,4)``plot(x(:,4),x(:,4).∗f(:,4),’m’)`

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**Figure 1.**Phase portrait views for the system (2).

**Figure 2.**Phase portrait views for system (3).

**Figure 16.**State by state phase portrait by (

**a**) vectors (left plots) and (

**b**) spline (right plots), for system (2).

**Figure 17.**State by state phase portrait by (

**a**) vectors (left plots) and (

**b**) spline (right plots), for system (3).

**Figure 18.**State by state phase portrait by (

**a**) vectors (left plots) and (

**b**) spline (right plots), for system (4).

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**MDPI and ACS Style**

Rodríguez-Licea, M.-A.; Perez-Pinal, F.-J.; Nuñez-Pérez, J.-C.; Sandoval-Ibarra, Y.
On the n-Dimensional Phase Portraits. *Appl. Sci.* **2019**, *9*, 872.
https://doi.org/10.3390/app9050872

**AMA Style**

Rodríguez-Licea M-A, Perez-Pinal F-J, Nuñez-Pérez J-C, Sandoval-Ibarra Y.
On the n-Dimensional Phase Portraits. *Applied Sciences*. 2019; 9(5):872.
https://doi.org/10.3390/app9050872

**Chicago/Turabian Style**

Rodríguez-Licea, Martín-Antonio, Francisco-J. Perez-Pinal, José-Cruz Nuñez-Pérez, and Yuma Sandoval-Ibarra.
2019. "On the n-Dimensional Phase Portraits" *Applied Sciences* 9, no. 5: 872.
https://doi.org/10.3390/app9050872