中文

A Beginner's Perspective on Dynamical Systems

Introduction to Dynamical Systems

Dynamical systems provide a mathematical framework to study how things change over time(dynamical). This blog aims to introduce the basic concepts and tools used to analyze 1-dimensional and 2-dimensional systems.

When I first encountered dynamical systems, I was incredibly confused. We already have Cartesian space, partial differential equations (PDEs), and ordinary differential equations (ODEs) to describe trajectories in Cartesian space—so why do we need phase space? Isn’t Cartesian space enough?

As I delved deeper into the subject, I realized the limitation of describing dynamics solely in Cartesian space: it lacks temporal information, and maybe this is where the word dynamical stands for.

Alt text describing the image

Consider the figure above, which shows a trajectory in 2D Cartesian space. A point moves from $(x_0,y_0)$ to $(x_1,y_1)$. While we can see the path the point takes, we have no way to tell how fast it moves along that path. The trajectory alone doesn’t reveal the system’s velocity or how it evolves over time.

This is where phase space becomes essential. By transitioning to phase space, we can visualize not only the position of the system but also its velocity (or momentum, depending on the context) at every point. For instance, in phase space, we can represent a system’s state using both $x$ and its velocity $ v=\dot{x} $ .This allows us to observe how the dynamics evolve over time with greater clarity.

Figures $ 1 $ and $2$ in phase space illustrate that two different dynamics can share the same trajectory in Cartesian space.

By visualizing the velocity at each position, phase space provides a richer, more complete picture of the system’s behavior. It becomes a powerful tool for studying how trajectories change, identifying patterns like fixed points, limit cycles, or chaotic behavior, and understanding the system’s overall dynamics.

1. One-Dimensional Flow

Let’s begin with a one-dimensional system. The first unfamiliar notation you’ll likely encounter in dynamical systems is $ \dot{x} $. But don’t worry—it’s simply another way of writing $\frac{dx}{dt}$. Typically, we express this as $ \dot{x}= f(x) $, which means that for every value of $x$, the system has a corresponding rate of change $\frac{dx}{dt}= f(x)$. For example, if $x$ represents position, then $ \dot{x}= f(x) $ describes the velocity at that specific position.

1.1 Fixed Points

1.2 Linear stability analysis

This is a very useful technique to expand the flow around fixed point, so that we can observe how higher order term dictates the dynamic around the fixed point. A general flow $ \dot{x} = f(x) $ can be solved close to any fixed point $ x = x^* $. A small deviation $ \eta(t) = x(t) - x^* $ from $ x^* $ evolves according to
\(\dot{\eta} = \dot{x} - \frac{d}{dt} x^* = \dot{x} = f(x)\)

Series expand the flow around the fixed point to linear order:
\(\dot{\eta} = f(x) = \underbrace{f(x^*)}_{=0} + f'(x^*) \underbrace{(x - x^*)}_{=\eta} + \frac{1}{2} f''(x^*) \underbrace{(x - x^*)^2}_{=\eta^2} + \cdots\)

\[\approx f'(x^*) \eta\]

where $f(x^*) = 0 $ ($ \frac{d}{dt}x^*=0 $) accordingto the definition of a fixed point. Solution: \(\eta = \eta_0 e^{f'(x^*)t}\)

stability exponent: $\lambda = f’(x^*)$ is the stability exponent,It determines whether the fixed point is stable (λ < 0) or unstable (λ > 0).

be careful! the stability exponent is not $\lambda = \ddot{x}$ . This is incorrect because it means$ \ddot{x} = \frac{d^2 x}{dt^2} $, while the stability exponent is the derivative of $ \dot{x} $ with respect to the coordinate $ x $, $ \lambda \equiv \frac{\partial \dot{x}}{\partial x} $.

1.3 Bifurcations in 1D

What is bifurcation? A bifurcation is a qualitative change in the dynamics of a system as a parameter is varied. Imagine a box resting on a pole. As the weight of the box changes, the curvature of the pole also changes to balance the system. However, if we keep increasing the weight of the box, there may come a point where the pole breaks. This breaking point represents a bifurcation in the box-pole system: beyond this point, the pole can no longer return to any stable curvature, regardless of the weight of the box. Alt text describing the image

2. Flow on the Plane

Flow in the plane is far more intriguing than in 1D, as it exhibits a richer global structure. Instead of a single fixed point, you can encounter phenomena such as closed orbits, limit cycles (isolated closed orbits), and homoclinic or heteroclinic orbits, among others.

2.1 Fixed Points in 2D

and the fixed points type can be classfied as below:

By referencing the ‘Zoo’ of fixed points, one can quickly determine the type of a fixed point using the trace and determinant of the Jacobian matrix. Alt text describing the image

For a 2D system \(\dot{x} = Ax \, , \, A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \, .\) according to the characteristic equation $0 = \det(A - \lambda I) = \lambda^2 - \tau\lambda + \Delta$( where $\tau = a + d$, namely the Trace of the matrix, and $\Delta = ad - bc$ which is the determinant of the matrix), we can get the eigenvalues of the system.

Why are we interested in the eigenvalues of the system? Because they allow us to transform the system so that its eigenvectors are aligned with the coordinate axes. This simplifies the analysis, as we can focus on how the eigenvalues dictate the system’s behavior near the fixed point. If the real part of an eigenvalue is greater than 0, the system expands exponentially along the corresponding eigenvector direction. Conversely, if the real part of an eigenvalue is less than 0, the system contracts exponentially along that direction.

The proof below demonstrates that eigenvalues act as the exponents in the solution trajectories, explaining the origins of exponential expansion and contraction.

$$ \dot{x}=Ax = PDP^{-1}x \\ \Rightarrow \frac{d}{dt}[P^{-1}x] = D \underbrace{P^{-1}x}_{\xi} \\\Rightarrow \dot{\xi} = D\xi \\ \Rightarrow \xi(t) = \begin{pmatrix} e^{\lambda_1 t} \xi_1(0) \\ e^{\lambda_2 t} \xi_2(0) \end{pmatrix}$$

When I first learned this, I was confused about why zero is the threshold and not one. I initially thought that repeatedly applying a matrix to a given vector would cause the vector to grow in the direction where $|\lambda|>1$ and shrink in the direction where $|\lambda|<1$.

My misunderstanding arose because the matrix is not directly applied to a physical entity but rather operates in the phase space, specifically on the ODE system. When solving the ODE system to obtain the solution, $\lambda$ represents the exponent rather than a multiplier!

And this is where oscillations come into play!!when eigenvalues are complex, the solution looks like this \(\xi(t) = \xi_1(0) e^{\mu t} \begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \end{pmatrix}\), where $\mu$ is the real part and $\omega$ is the complex part of the eigenvalues.

2.2 Non-Local Structures

Thus far, we have only discussed the local behavior of a system around a specific point, namely a fixed point. However, in 2D systems, new structures such as closed orbits and limit cycles can also emerge.

2.3 Bifurcations of Limit Cycles

2.4 Planar Bifurcations

Aside from the bifurcation of limit cycles in the phase plane, another type of bifurcation that can occur in 2D systems is the Hopf bifurcation.

2.5 Index Theory

Poincaré Index of a vector field in the plane is an integer that describes global information about the phase portrait around isolated singular points. The value of the index is determined by how the vector field orients around the singular points.

Conclusion

This blog covers only a tiny fraction of the field, which is both fascinating and incredibly challenging. It took me a significant amount of time to grasp these concepts and prepare for the exam! In the future, if I have the opportunity to work or study further in this field, I hope to explore and write about advanced topics such as chaos, strange attractors, fractals, and more.

Key takeaways

References

*****

☕ Happy learning journey~ 🛠️