Question

A generalization of the damped pendulum equation discussed in the text, or a damped spring-mass system, is the Liénard equation $$ \frac{d^{2} x}{d t^{2}}+c(x) \frac{d x}{d t}+g(x)=0 $$ If \(c(x)\) is a constant and \(g(x)=k x,\) then this equation has the form of the linear pen- \(\text { dulum equation [replace }\sin \theta \text { with } \theta \text { in Eq. ( } 12) \text { of Section } 9.2]\); otherwise the damping force \(c(x) d x / d t\) and restoring force \(g(x)\) are nonlinear. Assume that \(c\) is continuously differentiable, \(g\) is twice continuously differentiable, and \(g(0)=0 .\) (a) Write the Lienard equation as a system of two first equations by introducing the variable \(y=d x / d t\). (b) Show that \((0,0)\) is a and \(g\), then the critical point is asymptotically stable, (c) Show that if \(c(0) \geq 0\) and \(g^{\prime}(0)>0\), then the critical point is asymptotically stable, and that if \(c(0)<0\) or \(g^{\prime}(0)<0\), then the critical point is asymptotically stable, and that if \(c \text { ( } 0)<0\) or \(g^{\prime}(0)<0\), then the critical point is unstable of \(x=0\). Hint: Use Taylor series to approximate \(c\) and \(g\) in the neighborhood of \(x=0\)

Short answer

Question: Write the Liénard equation as a system of two first-order equations and discuss the properties of the critical point (0, 0) and its stability. Answer: The Liénard equation can be rewritten as a system of two first-order equations: $$\frac{dx}{dt} = y$$ $$\frac{dy}{dt} = -c(x) y - g(x)$$ At the critical point (0, 0), the eigenvalues of the Jacobian matrix determine its stability properties. If \(c(0) \geq 0\) and \(g^{\prime}(0) > 0\), the critical point is asymptotically stable. If \(c(0) < 0\) or \(g^{\prime}(0) < 0\), the critical point is unstable.

Step-by-step solution

Rewrite the Liénard equation as a system of two first-order equations

We will introduce a new variable \(y = \frac{dx}{dt}\). By differentiating \(x\) with respect to time \(t\), we have \(\frac{dy}{dt} = \frac{d^{2} x}{dt^{2}}\). Therefore, the Liénard equation can be rewritten as two first-order equations: $$\frac{dx}{dt} = y$$ $$\frac{dy}{dt} = -c(x) y - g(x)$$ The system of two first-order equations is now: $$\frac{dx}{dt} = y$$ $$\frac{dy}{dt} = -c(x) y - g(x)$$

Investigate the stability of the critical point\((0, 0)\)

The critical point \((0, 0)\) occurs when \(x = 0\) and \(y = 0\). According to the given conditions, \(g(0) = 0\). We will obtain the Jacobian matrix of the system at the critical point. The Jacobian matrix is given by: $$J(x, y) = \begin{pmatrix} \frac{\partial y}{\partial x} & \frac{\partial y}{\partial y} \\ \frac{\partial (-c(x)y - g(x))}{\partial x} & \frac{\partial (-c(x)y - g(x))}{\partial y} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -g^{\prime}(x) - y c^{\prime}(x) & -c(x) \end{pmatrix} $$ At the critical point \((0,0)\), the Jacobian matrix becomes: $$ J(0, 0) = \begin{pmatrix} 0 & 1 \\ -g^{\prime}(0) & -c(0) \end{pmatrix} $$

Analyze various cases of parameters

We will now show whether the critical point is asymptotically stable or unstable based on the initial conditions. Case 1: If \(c(0) \geq 0\) and \(g^{\prime}(0) > 0\), then the critical point is asymptotically stable. The eigenvalues of the Jacobian matrix at the critical point \((0, 0)\) are given by the characteristic equation:\(\lambda^2 + c(0) \lambda + g^{\prime}(0) = 0\). Since \(c(0) \geq 0\) and \(g^{\prime}(0) > 0\), the eigenvalues are both negative or have negative real parts, making the critical point asymptotically stable. Case 2: If \(c(0) < 0\) or \(g^{\prime}(0) < 0\), then the critical point is unstable. If \(c(0) < 0\) or \(g^{\prime}(0) < 0\), the eigenvalues of the Jacobian matrix at the critical point \((0, 0)\) will either have positive real parts or be complex with positive real parts, making the critical point unstable. In conclusion, we have rewritten the Liénard equation as a system of two first-order equations and analyzed the various cases of parameters to show whether the critical point \((0, 0)\) is asymptotically stable or unstable.

Key concepts

Damped Pendulum Equation

The concept of the damped pendulum is a crucial topic when studying dynamics and vibrations. It refers to a pendulum that experiences friction, often due to air resistance or internal mechanisms, which results in the gradual reduction of its amplitude over time. The mathematical representation of a damped pendulum can be described by the Liénard equation: \[\frac{d^{2} x}{d t^{2}}+c(x) \frac{d x}{d t}+g(x)=0 \]Where \(x\) denotes the displacement and \(t\) represents time. The term \(c(x) \frac{d x}{d t}\) is the damping force that is proportionate to the velocity and opposes the motion, causing the pendulum to eventually come to rest. The function \(g(x)\) acts as the restoring force, typically a function of displacement, that pulls the pendulum back toward its equilibrium position. In a typical linear damped pendulum where \(c(x)\) is constant and \(g(x) = kx\), the equation simplifies further, but in the general case of the Liénard equation, the forces may be nonlinear. Understanding this equation allows one to predict how the pendulum will behave over time and is an essential step in various engineering and physics calculations.

Spring-Mass System

Similarly, the spring-mass system is another fundamental concept in the study of oscillatory systems commonly encountered in physics and engineering. It consists of a mass attached to a spring, and when the mass is displaced from its equilibrium position, the spring exerts a force that tries to restore the system to equilibrium. This situation is also modeled by a form of the Liénard equation. If we let \(x\) denote the displacement of the mass from the equilibrium, and considering Hooke's Law where the restoring force is directly proportional to the displacement, we have \(g(x) = kx\). Adding a damping term, which is proportional to the velocity to represent any resistance (like air resistance or internal friction), we arrive at the damped harmonic oscillator equation which is a special case of the Liénard equation. For the spring-mass system, understanding the dynamic response and stability under various conditions such as changes in mass, spring stiffness, or damping coefficient, is critical for designing stable and efficient mechanical systems.

Stability Analysis

Stability analysis is an important aspect of system's theory, where one determines whether a system will return to equilibrium after a small disturbance. For the Liénard system, stability analysis focuses on determining the stability of the critical point. The critical point refers to the system's equilibrium state where the first derivative, which represents the velocity in our context, is zero. Stability analysis involves examining the eigenvalues of the Jacobian matrix at the critical point. If all eigenvalues have negative real parts, the critical point is considered asymptotically stable; small disturbances will decay over time, and the system will return to the equilibrium. If any eigenvalue has a positive real part, the system is deemed unstable; any disturbance will grow over time, and the system will not return to the equilibrium. Through stability analysis, one can predict long-term behavior of a system and ensure robust system design.

Critical Point

The critical point in a dynamical system is a point in the phase space at which the derivative of the system is zero—meaning that there is no change occurring at this point, and it represents a potential point of equilibrium. According to the Liénard equation, a critical point is found at the origin \((0, 0)\) where the displacement and its first derivative (considered as the velocity in mechanical systems) are both zero. Determining the nature of a critical point, whether it is stable or unstable, is essential for understanding the behavior of the system around that point. In a physical context, this could mean distinguishing between a system that will return to rest after being disturbed and one that will deviate even further from its initial position.

Jacobian Matrix

The Jacobian matrix is a fundamental tool in multiple fields, such as multivariable calculus, differential equations, and dynamical systems. It is a matrix of all first-order partial derivatives of a vector-valued function. In the context of the Liénard equation, the Jacobian matrix helps analyze the dynamics near the critical point. By evaluating the partial derivatives of the system's two first-order equations at the critical point, we can construct the Jacobian matrix as follows: \[J(x, y) =\begin{pmatrix}0 & 1 \-g^{\textprime}(x) - y c^{\textprime}(x) & -c(x)\end{pmatrix}\]The completed Jacobian matrix reveals the linear approximation of the dynamical system around the critical point, providing insight into the local behavior of the system.

Eigenvalues

Eigenvalues are a set of scalars associated with a system of linear equations or a matrix. They play a crucial role in stability analysis of dynamical systems. By determining the eigenvalues of the Jacobian matrix at the critical point, one can infer the stability of the system. Specifically, the eigenvalues are the roots of the characteristic equation derived from the Jacobian matrix, which for the Liénard equation are solutions to \[\lambda^2 + c(0) \lambda + g^{\textprime}(0) = 0\]In stability analysis, if the eigenvalues are real and negative, or have negative real parts in case of complex numbers, the system will be asymptotically stable at the critical point. If an eigenvalue is positive or has a positive real part, the system is unstable. Understanding the concept of eigenvalues is essential for engineers and physicists as it aids in designing efficiently stable systems.