Conditions For Non-Existence Of Periodic Orbits In Dynamical Systems

In the fascinating realm of dynamical systems, the existence and stability of periodic orbits play a pivotal role in understanding the long-term behavior of various phenomena. Periodic orbits, also known as limit cycles, represent self-sustained oscillations that arise in many physical, biological, and engineering systems. However, determining the conditions under which these periodic orbits do not exist is equally crucial for comprehending the dynamics of these systems. In this comprehensive article, we delve into the intricate world of ordinary differential equations and explore the conditions that guarantee the absence of periodic orbits in a specific dynamical system.

The Dynamical System and the Quest for Non-Existence

Consider the following two-dimensional autonomous system of ordinary differential equations:

$$\begin{aligned} x' &= y \\ y' &= ax - by - x^2y - x^3 \end{aligned}$$

where $x'$ and $y'$ denote the derivatives of $x$ and $y$ with respect to time, and $a$ and $b$ are real parameters. Our primary objective is to identify the conditions on $a$ and $b$ that ensure the non-existence of periodic orbits in this system. In other words, we seek to determine the values of $a$ and $b$ for which the system does not exhibit any self-sustained oscillations.

The Significance of Periodic Orbits

Before embarking on our quest, it is essential to appreciate the significance of periodic orbits in the broader context of dynamical systems. Periodic orbits represent repeating patterns in the system's behavior, where the state variables oscillate regularly over time. These oscillations can arise in diverse applications, ranging from the rhythmic beating of the heart to the cyclical fluctuations in predator-prey populations. Understanding the conditions that govern the existence and stability of periodic orbits is thus crucial for predicting and controlling the behavior of these systems.

The Challenge of Non-Existence

While determining the existence of periodic orbits can be a challenging task in itself, establishing their non-existence often poses an even greater hurdle. Unlike existence proofs, which typically rely on constructive arguments or fixed-point theorems, non-existence proofs often require more subtle and indirect approaches. These approaches may involve analyzing the system's energy dissipation properties, exploiting topological constraints, or employing specialized techniques like the Bendixson-Dulac criterion.

Initial Observations and the Case of Non-Positive 'a'

As a first step in our investigation, let us consider the scenario where the parameter $a$ is non-positive, i.e., $a \leq 0$. In this case, we can make some preliminary observations about the system's behavior.

The Role of 'a' in System Dynamics

The parameter $a$ plays a crucial role in determining the stability of the system's equilibrium points. Equilibrium points are those states where the system's dynamics come to a standstill, i.e., where both $x'$ and $y'$ are equal to zero. In our system, the equilibrium points are the solutions to the equations:

$$\begin{aligned} y &= 0 \\ ax - by - x^2y - x^3 &= 0 \end{aligned}$$

From the first equation, we have $y = 0$. Substituting this into the second equation, we get:

$$ax - x^3 = 0$$

Factoring out $x$, we obtain:

$$x(a - x^2) = 0$$

Thus, the equilibrium points are $x = 0$ and $x = \pm\sqrt{a}$. However, since we are considering the case where $a \leq 0$, the equilibrium points $x = \pm\sqrt{a}$ are only real-valued when $a = 0$. Therefore, when $a < 0$, the only equilibrium point is the origin $(0, 0)$.

The Origin as a Global Attractor

When $a < 0$, the origin $(0, 0)$ becomes a global attractor for the system. This means that all trajectories in the phase plane, regardless of their initial conditions, will eventually spiral into the origin as time progresses. The absence of other equilibrium points prevents the system from exhibiting any complex dynamics, such as oscillations or limit cycles. To rigorously prove this, we can employ Lyapunov stability theory, which provides a powerful framework for analyzing the stability of dynamical systems.

Lyapunov Stability Theory

Lyapunov stability theory centers around the concept of a Lyapunov function, a scalar function that serves as a measure of the system's energy or distance from an equilibrium point. If we can find a Lyapunov function that decreases along the system's trajectories, we can conclude that the equilibrium point is stable. Moreover, if the Lyapunov function decreases strictly along trajectories, we can establish asymptotic stability, meaning that the system's state will converge to the equilibrium point as time tends to infinity.

Constructing a Lyapunov Function

For our system, a suitable Lyapunov function candidate is:

$$V(x, y) = \frac{1}{2}(x^2 + y^2)$$

This function represents the square of the Euclidean distance from the origin. To check if $V(x, y)$ is a Lyapunov function, we need to examine its time derivative along the system's trajectories. The time derivative of $V(x, y)$ is given by:

$$\dot{V}(x, y) = \frac{\partial V}{\partial x}x' + \frac{\partial V}{\partial y}y' = x x' + y y'$$

Substituting the expressions for $x'$ and $y'$ from our system, we get:

$$\dot{V}(x, y) = x(y) + y(ax - by - x^2y - x^3) = xy + axy - by^2 - x^2y^2 - x^3y$$

Rearranging the terms, we have:

$$\dot{V}(x, y) = (1 + a)xy - by^2 - x^2y^2 - x^3y$$

When $a < 0$ and $b > 0$, we can see that $\dot{V}(x, y)$ is negative definite in a neighborhood of the origin. This means that $\dot{V}(x, y) < 0$ for all $(x, y)$ in that neighborhood, except for the origin itself, where $\dot{V}(0, 0) = 0$. Therefore, $V(x, y)$ is a Lyapunov function, and the origin is asymptotically stable.

Global Asymptotic Stability

Furthermore, we can show that the origin is globally asymptotically stable, meaning that it attracts all trajectories in the phase plane. To do this, we need to demonstrate that $\dot{V}(x, y) < 0$ for all $(x, y) \neq (0, 0)$.

We can rewrite $\dot{V}(x, y)$ as:

$$\dot{V}(x, y) = -by^2 + xy(1 + a - x^2) - x^2y^2$$

Since $a < 0$, we have $1 + a < 1$. Also, $x^2 \geq 0$, so $1 + a - x^2 < 1$. Therefore, if $y$ is sufficiently large, the term $-by^2$ will dominate, and $\dot{V}(x, y)$ will be negative. On the other hand, if $y$ is small, the term $-x^2y^2$ will dominate, and $\dot{V}(x, y)$ will also be negative.

Thus, we can conclude that $\dot{V}(x, y) < 0$ for all $(x, y) \neq (0, 0)$, and the origin is globally asymptotically stable. This implies that there are no periodic orbits in the system when $a < 0$ and $b > 0$, as all trajectories spiral into the origin.

The Bendixson-Dulac Criterion: A Powerful Tool for Non-Existence

Another powerful technique for proving the non-existence of periodic orbits is the Bendixson-Dulac criterion. This criterion provides a sufficient condition for the absence of closed trajectories in a two-dimensional dynamical system. It states that if there exists a continuously differentiable function $B(x, y)$ such that the expression

$$\frac{\partial}{\partial x}(B(x, y)f(x, y)) + \frac{\partial}{\partial y}(B(x, y)g(x, y))$$

has a fixed sign (either strictly positive or strictly negative) in a simply connected region of the phase plane, then the system has no periodic orbits in that region. Here, $f(x, y)$ and $g(x, y)$ are the right-hand sides of the system's differential equations.

Applying the Bendixson-Dulac Criterion

For our system, we have $f(x, y) = y$ and $g(x, y) = ax - by - x^2y - x^3$. Let us choose the Dulac function $B(x, y) = 1$. Then, we need to compute the expression:

$$\frac{\partial}{\partial x}(B(x, y)f(x, y)) + \frac{\partial}{\partial y}(B(x, y)g(x, y)) = \frac{\partial}{\partial x}(y) + \frac{\partial}{\partial y}(ax - by - x^2y - x^3)$$

Taking the partial derivatives, we get:

$$0 + (-b - x^2) = -b - x^2$$

If $b > 0$, then $-b - x^2 < 0$ for all $x$. Therefore, by the Bendixson-Dulac criterion, the system has no periodic orbits in the entire phase plane when $b > 0$.

Summarizing the Non-Existence Conditions

Combining our findings from Lyapunov stability theory and the Bendixson-Dulac criterion, we can conclude that the system

$$\begin{aligned} x' &= y \\ y' &= ax - by - x^2y - x^3 \end{aligned}$$

has no periodic orbits when $a \leq 0$ and $b > 0$. This provides a clear set of conditions on the parameters $a$ and $b$ that guarantee the absence of self-sustained oscillations in the system.

Further Exploration: The Case of Positive 'a'

While we have successfully identified conditions for the non-existence of periodic orbits when $a \leq 0$, the case of positive $a$ remains an open question. When $a > 0$, the system exhibits a richer dynamical behavior, with the possibility of multiple equilibrium points and more complex trajectories. Investigating the existence and non-existence of periodic orbits in this case requires more advanced techniques, such as the Poincaré-Bendixson theorem and bifurcation analysis.

The Poincaré-Bendixson Theorem

The Poincaré-Bendixson theorem is a fundamental result in the theory of two-dimensional dynamical systems. It provides a powerful tool for proving the existence of periodic orbits in certain situations. The theorem states that if a trajectory remains confined within a bounded region of the phase plane and does not approach any equilibrium points, then the trajectory must either be a periodic orbit or spiral towards a periodic orbit.

Bifurcation Analysis

Bifurcation analysis is a technique used to study how the qualitative behavior of a dynamical system changes as parameters are varied. In the context of periodic orbits, bifurcation analysis can reveal how the number and stability of limit cycles change as $a$ and $b$ are varied. This can provide valuable insights into the conditions under which periodic orbits appear or disappear.

The Complexity of Positive 'a'

Exploring the case of positive $a$ is beyond the scope of this article, but it serves as a reminder that the dynamics of even seemingly simple systems can be remarkably complex. The interplay between parameters, equilibrium points, and trajectory behavior can lead to a wide range of phenomena, including oscillations, chaos, and bifurcations.

Conclusion

In this article, we have embarked on a journey to uncover the conditions for the non-existence of periodic orbits in a specific dynamical system. By employing Lyapunov stability theory and the Bendixson-Dulac criterion, we have successfully established that the system

$$\begin{aligned} x' &= y \\ y' &= ax - by - x^2y - x^3 \end{aligned}$$

has no periodic orbits when $a \leq 0$ and $b > 0$. This result provides a clear understanding of how the parameters $a$ and $b$ influence the system's oscillatory behavior. While the case of positive $a$ presents a more challenging scenario, the techniques and concepts discussed here lay the foundation for further exploration into the fascinating world of dynamical systems.

The quest for understanding the existence and non-existence of periodic orbits is not merely an academic exercise. It has profound implications for various fields, including engineering, biology, and economics. By unraveling the intricate dynamics of these systems, we can gain valuable insights into the behavior of real-world phenomena and develop strategies for control and prediction. The journey continues, and the mysteries of dynamical systems await our exploration.