Limit Cycles

Recall

If you’re going through hell, keep on going, Winston Churchill.

Sometimes people don’t want to hear the truth because they don’t want their illusions destroyed, Friedrich Nietzsche.

Differential equations

Definition. A differential equation is an equation that involves one or more dependent variables, their derivatives with respect to one or more independent variables, and the independent variables themselves, e.g., $\frac{dy}{dx} = 3x +5y, y’ + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$

It involves (e.g., $\frac{dy}{dx} = 3x +5y$):

• Dependent variables: Variables that depend on one or more other variables (y).
• Independent variables: Variables upon which the dependent variables depend (x).
• Derivatives: Rates at which the dependent variables change with respect to the independent variables, $\frac{dy}{dx}$

First-Order Linear Ordinary Differential Equations (ODEs)

Definition. A first-order linear ordinary differential equation is an ordinary differential equation (ODE) involving an unknown function y(x), its first derivative y′, and functions of the independent variable x, which can be written in the general form:: a(x)y' + b(x)y = c(x) where:

• y = y(x) is the unknown function of the independent variable x.
• y′ = $\frac{dy}{dx}$ is the derivative of y with respect to x.
• a(x), b(x), and c(x) are known functions of x.

These equations are termed “linear” because the unknown function y and its derivative y’ appear to the first power and are not multiplied together or composed in any nonlinear way.

If the function c(x)=0 for all x in the interval of interest, the equation simplifies to: a(x)y’ + b(x)y = 0. Such an equation is called a homogeneous linear differential equation.

The Existence and Uniqueness Theorem provides crucial insight into the behavior of solutions to first-order differential equations ODEs. It states that if:

• The function f(x, y) (the right-hand side of the ODE) in the equation y’ = f(x, y) is continuous in some neighborhood around a point (x₀, y₀) and
• Its partial derivative with respect to y, denoted as $\frac{∂f}{∂y}$, is also continuous near (x₀, y₀).

Then, the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x₀, y₀), meaning that it satisfies the initial condition y(x₀) = y₀.

This theorem ensures that under these conditions, the solution exists and is unique near x = x₀.

Limit Cycles in Non-Linear Autonomous System

In the study of differential equations and dynamical systems, non-linear autonomous systems are of great interest and play a crucial role due to their complex and rich behavior. One of the most fascinating phenomena exhibited by such systems is the occurrence of limit cycles.

A limit cycle is a closed, isolated trajectory in the phase plane that represents a periodic solution.

We consider a general two-dimensional non-linear autonomous system of the form:

$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$ where:

• x’ and y’ denote the time derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$, respectively.
• f(x, y) and g(x, y) are non-linear functions that govern the time evolution of the variables x and y. This system describes how the state (x, y) evolves over time, based on their current values.

By “autonomous,” we mean that these functions do not explicitly depend on time. The evolution of the system depends only on the current state (x, y), not on what time it is.

Visualizing the System: The Velocity Field

To gain a better understanding of the behavior of this system, we can represent the system as a velocity (or vector) field $\vec{F}$ in the plane. The velocity field is defined as: $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where

• $f(x, y)\hat{\mathbf{i}}$ represents the velocity component in the x-direction.
• $g(x, y)\hat{\mathbf{j}}$ represents the velocity component in the y-direction.
• $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ are unit vectors in the x and y axes, respectively.

This vector field describes how the values of x and y change over time at every point in the plane. Each point (x, y) has an associated vector $\vec{F}(x, y)$ indicating the instantaneous direction and speed of the system’s evolution starting from that point.

Thus, if we think of (x(t), y(t)) as the position of a “particle” moving in the plane, $\vec{F}(x, y)$ tells us the velocity of that particle at that instant.

Trajectories in the Phase Plane

Solutions to the system are pairs of functions x(t) and y(t), but geometrically, they are trajectories or paths traced out by the evolving system in the xy-plane (phase plane). These trajectories follow the direction of the vector field $\vec{F}$, meaning that at any point along the trajectory, the tangent to the path is given by $\vec{F}(x, y)$.

Refer to Figure i for a visual representation and aid in understanding it. Different initial conditions may lead to different trajectories, but these trajectories cannot cross each other. Non-crossing is a consequence of uniqueness of solutions: if two distinct solutions crossed at a point, that point would have two different future evolutions, which is impossible under standard conditions.

Critical Points: Where the Velocity Field Vanishes

A critical point of the system occurs where the velocity field is zero, i.e., where the system's rate of change is zero. Mathematically, this happens when both $f(x_0, y_0) = 0, g(x_0, y_0) = 0$, meaning that the velocity at the point (x₀, y₀) is zero. At such points, the system is at equilibrium, meaning there is no motion, and the solution remains constant over time.

In terms of the field, these points are where: $\vec{F}(x_0, y_0) = f(x_0, y_0)\hat{\mathbf{i}}+g(x_0, y_0)\hat{\mathbf{j}} = \vec{0}$

Critical points are important because they often represent stable or unstable equilibrium states where the system tends to “settle” or from which it may “escape” in the long term, depending on the nature of the equilibrium.

Stability of Critical Points

The behavior near critical points can be analyzed using linearization and the eigenvalues of the Jacobian matrix:

J = $(\begin{matrix}\frac{∂f}{∂x} & \frac{∂f}{∂y}\\ \frac{∂g}{∂x} & \frac{∂g}{∂y}\end{matrix})\bigg|_{(x_0, y_0)}$

The eigenvalues of J determine the local behavior near the critical point:

• If both eigenvalues have negative real parts, the critical point is a stable node or stable focus. Solutions or trajectories near this critical point converge to it as t → ∞.
• If both eigenvalues have positive real parts, it’s an unstable node or unstable focus. Solutions or trajectories near this critical point diverge or move away from it as t → ∞.
• If eigenvalues are real and of opposite signs, the critical point is a saddle point. Solutions approach the critical point along one eigenvector and move away along the other, making it unstable.
• If eigenvalues are purely imaginary conjugates (with zero real part), nearby solutions or trajectories circle around the equilibrium. The critical point is a center. Solutions orbit around the critical point without converging or diverging.

Closed Trajectories and Limit Cycles

A closed trajectory is a path in the xy-plane that loops back onto itself (to its starting point) and then repeats itself indefinitely. In other words, if a trajectory is closed, the system exhibits periodic behavior, that is, after a certain period T, the system returns to its initial state, and this cycle repeats over and over.

Geometrically, a closed trajectory represents a periodic solution (x(t), y(t)) such that: x(t + T) = x(t), y(t + T) = y(t) for all t.

Importantly, a closed trajectory does not cross itself —the system cannot have two different velocity directions at the same point due to the uniqueness of solutions in differential equations (Refer to Figure v for a visual representation and aid in understanding it). This property ensures that the system’s motion is uniquely determined by the initial conditions and evolves in a smooth, continuous manner.

Example: A Simple Harmonic Oscillator

The simple harmonic oscillator is a fundamental example in differential equations, classical mechanics, and physics, representing systems that exhibit purely periodic motion, such as springs and pendulums under ideal conditions.

Common real-world examples include a mass attached to a spring (obeying Hooke’s law) or a small-angle pendulum, each oscillating around an equilibrium position. The simple harmonic oscillator is linear, conservative (no energy loss), and its solutions are sinusoidal, indicating perfect periodic motion.

Consider the system of differential equations: $\begin{cases} x’ = y \\ y’ = -x \end{cases}$ where x = x(t) and y = y(t) are functions of time t (two variables that oscillate over time), and x’ and y’ denote derivatives of x and y, respectively with respect to time. This system describes how the variables x and y evolve over time, with each variable depending on the other.

Combining these, we can also derive a second-order differential equation for x alone: x’’ = y’ = -x, which is the classic form of a simple harmonic oscillator equation. In other words, this system describes a simple harmonic oscillator.

The matrix representation of the system is: $\vec{x’} = Ax$ where A = $(\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix})$ is the coefficient matrix, and $\vec{x} = (\begin{smallmatrix}x\\ y\end{smallmatrix})$ is the state vector. This compact form allows us to use linear algebra techniques to solve the system.

To find the eigenvalues, we solve the characteristic equation: det(A-λI) = $det(\begin{smallmatrix}-λ & 1\\ -1 & -λ\end{smallmatrix}) = 0 ↭ (−λ)(−λ)−(1)(−1) = 0 ↭ λ^2 + 1 = 0 ↭ λ^2 = -1$. The eigenvalues are λ₁ = i and λ₂ = -i. The eigenvalues are purely imaginary. This indicates that the system exhibits rotational behavior (oscillatory) in the xy-plane (phase plane), trajectories in the (x, y)-plane will rotate around the origin, never spiraling in or out.

To find the eigenvectors, we substitute each eigenvalue back into $(A−λI)\vec{v} = \vec{0}$

For λ₁ = i, $(\begin{smallmatrix}-i & 1\\ -1 & -i\end{smallmatrix})(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) = (\begin{smallmatrix}0\\ 0\end{smallmatrix})$

From the first equation: -ia₁ +a₂ = 0 ⇒ a₂ = ia₁. From the second row: -a₁ -ia₂ = 0 ⇒[Substitute a₂ = ia₁] -a₁ -i(ia₁) = -a₁ + a₁ = 0. We can choose a₁ = 1 (since eigenvectors are determined up to a scalar multiple). Therefore, the eigenvector is: $\vec{v_1}=(\begin{smallmatrix}1\\ i\end{smallmatrix})$

For λ₂ = -i, $(\begin{smallmatrix}i & 1\\ -1 & i\end{smallmatrix})(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) = (\begin{smallmatrix}0\\ 0\end{smallmatrix})$

From the first equation: ia₁ +a₂ = 0 ⇒ a₂ = -ia₁. Choosing a₁ = 1, we have: $\vec{v_2}=(\begin{smallmatrix}1\\ -i\end{smallmatrix})$

The general complex solution to the system in complex form is a linear combination of the eigenvectors multiplied by the exponential of their eigenvalues: $\vec{x}(t) = c_1e^{it}\vec{v_1} + c_2e^{-it}\vec{v_2}$ where c₁ and c₂ are complex constants determined by initial conditions.

Substituting the eigenvectors: $\vec{x}(t) = c_1e^{it}(\begin{smallmatrix}1\\ i\end{smallmatrix})+c_2e^{-it}(\begin{smallmatrix}1\\ -i\end{smallmatrix})$

To find real-valued solutions, we express the complex exponentials using Euler’s formula: $e^{it} = cos(t) + isin(t), e^{-it} = cos(t) - isin(t)$

Substitute back into the general solution: $\vec{x}(t) = c_1(cos(t)+isin(t))(\begin{matrix}1\\ i\end{matrix})+c_2(cos(t)-isin(t))(\begin{matrix}1\\ -i\end{matrix}) = (\begin{matrix}c_1(cos(t)+isin(t))+c_2(cos(t)-isin(t))\\ c_1(icos(t)-sin(t))+c_2(-icos(t)+sin(t))\end{matrix})$

Let’s use the real and imaginary parts of the complex solutions to construct real solutions.

First Real Solution. Take the real part of $e^{it}\vec{v_1}$: $\mathbb{R}[e^{it}\vec{v_1}] = (\begin{smallmatrix}cos(t)\\ -sin(t)\end{smallmatrix})$

Second Real Solution. Take the imaginary part of $e^{it}\vec{v_1}$: $\mathbb{ℑ}[e^{it}\vec{v_1}] = (\begin{smallmatrix}sin(t)\\ cos(t)\end{smallmatrix})$

Therefore, the general real solution is: $c_1(\begin{smallmatrix}cos(t)\\ -sin(t)\end{smallmatrix})+c_2(\begin{smallmatrix}sin(t)\\ cos(t)\end{smallmatrix})$ where c₁ and c₂ are real constants determined by initial conditions.

The solutions x(t) and y(t) represent sinusoidal functions with the same frequency but possibly different amplitudes and phases. The trajectories in the phase plane are closed curves, specifically circles or ellipses, depending on the constants c₁ and c₂.

For the simple harmonic oscillator, this is a family of concentric circles centered at the origin, representing periodic motion [1]. Each trajectory is a closed curve, and the motion goes around clockwise indefinitely, which is typical of simple harmonic oscillators. (Refer to Figure ii for a visual representation and aid in understanding it).

In linear systems like the simple harmonic oscillator, all trajectories are closed curves (circles) but are not isolated —there is a family of closed trajectories filling the phase plane.

Linear systems do not have limit cycles in the strict sense because limit cycles are a feature of non-linear systems where closed trajectories are isolated.

Circular Trajectories

For the simple harmonic oscillator, the trajectories are circles centered at the origin [1]. To see this, consider the expressions x(t) = c₁cos(t) + c₂sin(t), y(t) = -c₁sin(t) + c₂cos(t).

$x^2(t)+y^2(t) =[\text{Expanding both terms}] c_1^2·cos^2(t) + 2c_1c_2sin(t)cos(t) + c_2^2·sin^2(t) + c_1^2·sin^2(t) -2c_1c_2sin(t)cos(t) +c_2^2cos^2(t) =[\text{Simplify}] c_1^2(cos^2(t)+sin^2(t)) + c_2^2(sin^2(t) + cos^2(t)) + (2c_1c_2sin(t)cos(t)-2c_1c_2sin(t)cos(t)) = (c_1^2+c_2^2)(cos^2(t)+sin^2(t)) = c_1^2+c_2^2 ↭ x^2(t)+y^2(t) = c_1^2+c_2^2 = R^2$ where R = $\sqrt{c_1^2+c_2^2}$ is the radius of the circle. Therefore, the trajectories of the system are circles of radius R centered at the origin.

Limit Cycles

A limit cycle is a special type of closed trajectory in the phase plane of a dynamical system. Consider a system described by two variables, x and y, whose values change over time according to a set of differential equations. The phase plane is a two-dimensional plane where the horizontal axis represents the variable x and the vertical axis represents the variable y. Each point in the phase plane corresponds to a particular state of the system (a particular combination of x and y).

A trajectory in the phase plane is the path traced out by the system’s state as it evolves over time from a given initial condition (x₀, y₀). A closed trajectory is one where the system eventually returns to the exact same state it started from, causing the path to form a closed loop.

A limit cycle is a closed trajectory in the phase plane that is isolated, meaning that nearby trajectories are not closed and either spiral towards or away from the limit cycle.

Limit cycles are significant because they represent sustained, ongoing oscillations in the system, which can be:

• A stable limit cycle is one where all neighboring trajectories approach the cycle as t → ∞. The system eventually returns to the limit cycle even if disturbed. In other words, a limit cycle attracts nearby trajectories, causing the system to settle into a repeating, periodic behavior. Nearby trajectories spiral inward toward the limit cycle, getting closer and closer, but never quite touching it. (Refer to Figure iii for a visual representation and aid in understanding it).
• An unstable limit cycle repels nearby trajectories. In this case, any small disturbance will cause the system to move away from the limit cycle as time progresses. Thus, neighboring trajectories diverge from the limit cycle as t → ∞. The system does not return to the periodic motion described by the limit cycle after a disturbance because this periodic motion is not robust against perturbations.
• A semi-stable is one where trajectories on one side (direction) approach the limit cycle, while those on the other side move away. This means the limit cycle is stable from one direction and unstable from the other. This creates a more complex stability scenario, neither fully stable nor fully unstable.

Limit cycles do not occur in linear systems; they are unique to non-linear systems due to their complex interactions. In linear systems, any closed trajectories (such as circles or ellipses generated by a center equilibrium) are not isolated —they form a continuum of closed orbits filling a region in the phase plane, and thus, they do not satisfy the definition of a limit cycle.

One real-world example of a limit cycle is the natural process of breathing. Breathing is a periodic motion that can be modeled as a limit cycle. If the system (your breathing) is disturbed, say, by a temporary obstruction, physical exercise, or a moment of stress, anxiety or panic, it will gradually return to its original, stable pattern of breathing. This resilience to disturbances and return to the original periodic motion is a hallmark, a key characteristic of a stable limit cycle.

The Existence Problem of Limit Cycles

Understanding and determining whether a given system has a limit cycle is a crucial challenge in the study of non-linear dynamical systems. Unfortunately, there is no universal formula or method to directly predict or determine the existence of limit cycles for every non-linear system.

There are various approaches used by mathematicians and researchers to predict and identify limit cycles:

1. Intuition and Physical Insight: Systems modeled after physical phenomena can be analyzed using intuition about these phenomena. For example, in ecological models like predator-prey dynamics or certain chemical reactions, oscillatory behavior is expected, suggesting the presence of limit cycles.
2. Computer Simulations. Since there are no universal analytical methods for finding limit cycles, scientists frequently rely on numerical simulations. Numerical simulations allow visualization of trajectories in the phase plane. Researchers simulate the system’s trajectories from various initial conditions. If a trajectory settles into a closed loop, this loop is likely a stable limit cycle. Software tools such as MATLAB, Mathematica, or Python libraries (like NumPy and SciPy) are commonly used.
3. Analytical Methods (partial tools): Although general methods are scarce, there are some analytical tools, like the Poincaré-Bendixson Theorem and Bendixson’s Criterion, that provide useful insights into the existence or non-existence of limit cycles in specific systems and scenarios.

While finding limit cycles can be difficult, in some cases, it is possible to rule out their existence by applying specific criteria.

Bendixson’s Criterion

Bendixson’s Criterion is a method for excluding the existence of closed trajectories (and hence, limit cycles) within a region of the plane.

Bendixson’s criterion. Let D be a simply connected region of the xy-plane. Consider a continuously differentiable vector field: $\vec{F} = f(x, y)\hat{\mathbf{i}} + g(x, y)\hat{\mathbf{j}}$ governing the system:

$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$

The divergence of the vector field is given by: $div \vec{F} = f_x + g_y = \frac{∂f}{∂x} + \frac{∂g}{∂y}$, where f(x, y) and g(x, y) are the components of the vector field governing the time evolution of x and y, respectively. If the divergence $div(\vec{F})$ is continuous throughout the region D and does not change sign (meaning it is always positive or always negative) and is not identically zero, then there are no closed trajectories (and therefore no limit cycles) lying entirely within D.

If the divergence of the vector field is always positive or always negative in the region D, it implies that the flow is either consistently diverging (spreading out, pushing trajectories outward behaving like a source) or converging (coming together, pulling trajectories inward behaving like a sink) throughout the region. This consistent behavior prevents the trajectories from closing onto themselves (loop around and close around) to form limit cycles within D.

Solved exercises

• Consider the system:

$\begin{cases} x’ = x^3 + y^3 \\ y’ = 3x + y^3 + 2y \end{cases}$

The vector field for this system is $\vec{F} = (x^3 + y^3)\hat{\mathbf{i}} + (3x + y^3 + 2y)\hat{\mathbf{j}}$

Let’s compute the divergence of the vector field: $div \vec{F} = f_x + g_y = 3x^2 + 3y^2 +2$. Notice that $div \vec{F} > 0$ in ℝ² (it is always positive since x²≥0, y²≥0 ∀x, y and there is an additional positive term 2)⇒[By Bendixson’s criterion] Since the divergence is always positive and constant throughout the plane, there can be no closed trajectories in the xy-plane ⇒There are no limit cycles anywhere in the system.

Important Caveat of Bendixson’s Criterion.

It’s important to note that Bendixson’s Criterion can only be used to exclude the possibility of limit cycles within a region. If the divergence changes sign or is zero somewhere in D, the criterion does not provide any information about the existence of limit cycles; other methods must be used.

Proof (indirect proof)

Assume, for the sake of contradiction, that there exists a closed trajectory C within the region D. Let R be the region enclosed by the curve C (Refer to Figure iv for a visual representation and aid in understanding it).

Consider the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}} + g(x, y)\hat{\mathbf{j}}$ governing the system:

$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$

Since C is a trajectory of the system, the vector field (velocity field) $\vec{F}$ is always tangent to the curve (the curve is a trajectory, it is supposed to be going in the direction given by the vector field) at every point. Therefore, the normal vector and the vector field are perpendicular, i.e., their dot product is zero: $\vec{F}·\hat{\mathbf{n}} = 0$.

The flux integral of the vector field $\vec{F}$ across C is given by:

$\oint_{C} \vec{F}·\hat{\mathbf{n}}ds$ where:

• $\hat{\mathbf{n}}$ is the outward-pointing unit normal vector to C.
• ds is the differential arc length along C.

As we have previously stated, since $\vec{F}$ is tangent to C, $\vec{F}·\hat{\mathbf{n}} = 0$. Therefore, the flux across C is zero: $\oint_{C} \vec{F}·\hat{\mathbf{n}}ds = 0$

By Green’s Theorem, the flux integral over a closed curve C can be converted to a double integral over the region R enclosed by C: $\oint_{C} \vec{F}·\hat{\mathbf{n}}dS = \int\int_{R} div(\vec{F})dA$, where:

• $div(\vec{F}) = \frac{∂f}{∂x}+\frac{∂g}{∂y}$ is the divergence of the vector field.
• dA is the differential area element.

Since the flux across C is zero, we have: $\int\int_{R} div(\vec{F})dA = 0$

However, we have already assumed that:

1. The divergence $div(\vec{F})$ is continuous in R.
2. $div(\vec{F})$ is not zero and does not change sign throughout R, meaning it is either strictly positive or strictly negative.

Under these conditions, the divergence is either $div(\vec{F}) > 0$ or $div(\vec{F}) < 0$ for all points in R. Since R has a positive area, and $div(\vec{F}) $ is non-zero of a constant sign the integral of the divergence over the region R $\int\int_{R} div(\vec{F})dA$ must be either strictly positive ($div(\vec{F}) > 0$) or negative ($div(\vec{F}) < 0$) ⊥ Therefore, the assumption that a closed trajectory C exists is false. Hence, under the given conditions, there are no closed trajectories (limit cycles) within the region D.

Critical point criterion

Critical point criterion (Poincaré-Bendixson) is closely related to the Poincaré-Bendixson theorem. If a closed trajectory C exists within a region D in the xy-plane, then there must be at least one critical point (also known as an equilibrium point) inside the region enclosed by that closed trajectory C.

A critical point is a point (x₀, y₀) where the system’s dynamic “pause”; in other words, the velocity vector is zero at that point. Formally, for a system:

$\begin{cases} x’ = f(x, y) \\ y’ = g(x, y) \end{cases}$

a critical point is a solution (x₀, y₀) satisfying f(x₀, y₀) = 0 and g(x₀, y₀) = 0.

Contrapositive Logic. The contrapositive of a statement “If A, then B” (A⇒B) is the logically equivalent statement “If not B, then not A” (¬B⇒¬A). Applying this to the critical point criterion:

• Original statement: If a closed trajectory exists, then there is a critical point inside it.
• Contrapositive: If there are no critical points in a region, then there are no closed trajectories (no limit cycles) in that region.

This means that if you can identify a region D in the plane that contains no critical points, then you can conclude that no closed trajectories (limit cycles) exist within D.

Does this system has limit cycles?

Consider the following non-linear autonomous system:

$\begin{cases} x’ = x^2 + y^2 + 1 \\ y’ = x^2 -y^2 \end{cases}$

To investigate whether limit cycles exist, we can first apply Bendixson’s Criterion. It states that that if the divergence of the vector field is continuous and does not change sign (i.e., it is either strictly positive or strictly negative) in a simply connected region D, then there are no closed trajectories (and hence no limit cycles) lying entirely within D.

Given the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where $f(x, y) = x^2 + y^2 + 1, g(x, y) = x^2 -y^2$. The divergence of the vector field $\vec{F}$ is given by: $div(\vec{F}) = \frac{∂f}{∂x} + \frac{∂g}{∂y} = 2x-2y$.

Analyzing the Divergence. Set the divergence equal to zero:   2x -2y = 0 ⇒ x -y = 0 ⇒ x = y. This tells us that along the line y = x, the divergence is zero.

Signs of the Divergence

• To the right of this line y = x (i.e., where x > y), the divergence is positive ($div(\vec{F}) = 2x−2y > 0$).
• To the left of this line (i.e., where x < y), the divergence is negative ($div(\vec{F}) =2x−2y < 0$).

Applying Bendixson’s Criterion

In regions where the divergence is strictly positive or strictly negative and continuous, Bendixson’s Criterion tells us that no closed trajectories can exist entirely within those regions.

• To the right of y = x: No closed trajectories exist.
• To the left of y = x: No closed trajectories exist. However, along the line y = x, the divergence is zero. Therefore, Bendixson’s Criterion does not rule out the possibility of closed trajectories that cross this line or lie along it.

Next, we use the critical point criterion to check whether the system has any critical points. The Critical Point Criterion states:

• If a closed trajectory exists, there must be at least one critical point (equilibrium point) inside the region enclosed by the trajectory.
• Contrapositive: If there are no critical points inside a closed region, then there are no closed trajectories entirely within that region.

Critical points occur where both derivatives x′ and y′are zero simultaneously.

However, this is impossible because: $x’ = x^2+y^2+1 > 0$ (there are not real solutions to x² + y² = -1) ⇒ There are no critical points in the real plane system. ⇒Since there are no critical points in the system, by the contrapositive of the critical point criterion, we can conclude that the system does not have any limit cycles.

Since $x’ > 0$ for all $(x, y)$, this implies all trajectories will move rightward indefinitely as time progresses, so they cannot look back on themselves to form a closed trajectory. This alone is sufficient to rule out limit cycles.

• Consider the spring Mass damper system with a mass of one (m = 1), and stiffness of 1 (k = 1), and a non-linear damping c (c = c(x) > 0).

To determine whether limit cycles can exist in the given spring-mass-damper system with non-linear damping, we will apply Bendixson’s Criterion. This criterion helps us identify whether closed trajectories, and hence limit cycles, are possible in a particular region of the phase plane by examining the divergence of the vector field.

The governing equation for this system is: $x’’ + c(x)x’ + x = 0.$

To analyze this system in the phase plane, we convert this into a first-order system by introduction a new variable y’ = x. Then, we have:

$\begin{cases} x’ = y \\ y’ = -x -c(x)y \end{cases}$

This gives us the vector field $\vec{F} = f(x, y)\hat{\mathbf{i}}+g(x, y)\hat{\mathbf{j}}$ where where f(x, y) = y, g(x, y) = -x -c(x)y.

To investigate whether limit cycles exist, we can apply Bendixson’s Criterion. It states that if the divergence of the vector field $\vec{F}$ is continuous and does not change sign (i.e., it is either strictly positive or strictly negative) in a simply connected region D, then there are no closed trajectories (and thus no limit cycles) can exist entirely within D.

The divergence of the vector field $\vec{F}$ is given by: $div(\vec{F}) = \frac{∂f}{∂x} + \frac{∂g}{∂y} = 0 -c(x)$

Applying Bendixson’s Criterion:

Since c(x) > 0 by definition, we have: $div(\vec{F}) = -c(x) < 0$ ∀x, y.

The divergence is strictly negative for all x and y (assuming c(x) is continuous). Thus:

• The divergence does not change sign in any region of the xy-plane.
• The divergence is also continuous in the phase plane since c(x) is continuous.
• By Bendixson’s Criterion, since the divergence is strictly negative and continuous across the entire plane, no closed trajectories (and therefore no limit cycles) can exist in this system.

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7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare [18.03 Differential Equations, Spring 2006], YouTube by MIT OpenCourseWare.
9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.