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Chapter 8: Problem 5

Prove that at any zero-eigenvalue bifurcation in two dimensions, the nullclines always intersect tangentially. (Hint: Consider the geometrical meaning of the rows in the Jacobian matrix.)

Short Answer

Expert verified
At any zero-eigenvalue bifurcation in two dimensions, considering the system of equations \(\frac{dx}{dt} = F(x, y)\) and \(\frac{dy}{dt} = G(x, y)\), the Jacobian matrix is given by \(J(x, y) = \begin{bmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} \end{bmatrix}\). Setting \(\lambda = 0\) in the characteristic equation, we find the relationship \(\frac{\partial F}{\partial x}\frac{\partial G}{\partial y} = \frac{\partial F}{\partial y}\frac{\partial G}{\partial x}\), which implies that the tangent vectors of the nullclines are parallel, proving that the nullclines intersect tangentially at the zero-eigenvalue bifurcation.

Step by step solution

01

Understand Zero-eigenvalue Bifurcation and Nullclines

A zero-eigenvalue bifurcation is a point where a system of differential equations undergoes a change in stability, which can be characterized by having a zero eigenvalue in the Jacobian matrix. Nullclines are curves in a phase plane where one of the variables has a zero derivative, i.e., the system remains constant along the nullcline.
02

Provide the System of Equations and Define the Jacobian Matrix

Consider a two-dimensional system of differential equations: \[\begin{cases} \frac{dx}{dt} = F(x, y)\\ \frac{dy}{dt} = G(x, y) \end{cases}\] The Jacobian matrix of this system is given by: \[J(x, y) = \begin{bmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} \end{bmatrix}\]
03

Identify the Eigenvalues of the Jacobian Matrix

Find the characteristic equation of the Jacobian matrix, which is given by the determinant: \[\det(J - \lambda I) = \begin{vmatrix} \frac{\partial F}{\partial x} - \lambda & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} - \lambda \end{vmatrix} \] Compute the determinant: \[(\frac{\partial F}{\partial x}- \lambda)(\frac{\partial G}{\partial y} - \lambda) - (\frac{\partial F}{\partial y})(\frac{\partial G}{\partial x}) = \lambda^2 - (\frac{\partial F}{\partial x}+\frac{\partial G}{\partial y})\lambda + \frac{\partial F}{\partial x}\frac{\partial G}{\partial y} - \frac{\partial F}{\partial y}\frac{\partial G}{\partial x} = 0\] At a zero-eigenvalue bifurcation, one of the eigenvalues of the Jacobian matrix is zero; therefore, we can set \(\lambda = 0\) in our characteristic equation.
04

Analyze the Relationship Between Rows of the Jacobian Matrix and Tangential Intersection

When \(\lambda = 0\), we have: \[0 = \frac{\partial F}{\partial x}\frac{\partial G}{\partial y} - \frac{\partial F}{\partial y}\frac{\partial G}{\partial x}\] which can be rewritten as: \[\frac{\partial F}{\partial x}\frac{\partial G}{\partial y} = \frac{\partial F}{\partial y}\frac{\partial G}{\partial x}\] The left-hand side of this equation represents the part of the tangent vector related to the \(x\)-nullcline, while the right-hand side represents the part related to the \(y\)-nullcline. This equation shows that these two tangent vectors are proportional, which implies that they are parallel. Since nullclines are defined by the condition that one variable has a zero partial derivative along its curve, we can conclude that, at any zero-eigenvalue bifurcation, the tangent vectors of the nullclines are parallel. This implies that the nullclines always intersect tangentially at a zero-eigenvalue bifurcation in two dimensions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nullclines
Nullclines are essential geometric tools used in phase plane analysis of dynamical systems. They are defined as curves where one of the derivatives of the variables in the system becomes zero. For a two-dimensional system, there are typically two nullclines: the \( x \)-nullcline, where \( \frac{dx}{dt} = 0 \), and the \( y \)-nullcline, where \( \frac{dy}{dt} = 0 \). These curves help us understand the behavior of the system without needing to solve the differential equations explicitly.
Nullclines are particularly useful because at their intersection points, the system is at equilibrium; both derivatives are zero, indicating no change in the system's state. This information can guide us in predicting the stability of equilibria in the system. By analyzing nullclines, we can easily visualize how changes in parameters or initial conditions can affect the system. This is crucial when studying complex behaviors such as bifurcations, where the stability of equilibria changes as parameters vary.
Jacobian Matrix
The Jacobian matrix is a fundamental tool in analyzing the stability of a dynamical system near equilibrium points. It is essentially the matrix of all first-order partial derivatives of the system functions, which provides a linear approximation of the system around an equilibrium. For our two-dimensional system defined by equations \( F(x, y) \) and \( G(x, y) \), the Jacobian is given by:\[J(x, y) = \begin{bmatrix}\frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \\frac{\partial G}{\partial x} & \frac{\partial G}{\partial y}\end{bmatrix}\]
By evaluating the Jacobian at equilibrium points, we can derive eigenvalues which inform us about the system's behavior near these equilibria. If the eigenvalues consist of a zero-eigenvalue, it indicates a point of bifurcation, suggesting a shift in the stability or nature of the equilibrium. The matrix can reveal how small perturbations impact the future states of the system. Understanding the Jacobian is crucial in differentiating between stable and unstable equilibria, making it a cornerstone concept in the study of differential equations and dynamical systems.
Phase Plane Analysis
Phase plane analysis is a technique used to graphically represent the trajectories of a dynamical system in a two-dimensional space. Each point in the phase plane corresponds to a specific state of the system, and trajectories show how the state evolves over time.
In the context of nullcline analysis, the phase plane is an effective way to visualize and understand equilibrium points and the corresponding stability of these points. By plotting trajectories alongside nullclines and equilibrium points, one can quickly identify behaviors such as limit cycles, fixed points, and spirals. MATLAB, Python, and similar software tools can be employed to aid in rendering these plots, providing a valuable visual representation.
Phase plane diagrams allow for the qualitative analysis of systems, making it easier to foresee the long-term behavior of a system given different initial conditions. This method can simplify the understanding of complex differential equations without requiring precise numerical solutions.
Tangential Intersection
Tangential intersection refers to the situation where two curves touch but do not cross each other at the point of intersection. In the context of bifurcation theory and nullclines, a tangential intersection occurs when the tangent lines of the curves at the intersection point are parallel.
This concept becomes particularly significant during zero-eigenvalue bifurcations in two-dimensional systems. As explained, the condition given is:\[\frac{\partial F}{\partial x}\frac{\partial G}{\partial y} = \frac{\partial F}{\partial y}\frac{\partial G}{\partial x}\]This shows that the gradients of the nullclines, which are determined by the rows of the Jacobian matrix, are proportional, implying a tangential intersection.
At a zero-eigenvalue bifurcation, the system's stability characteristics undergo changes, and the tangential intersection of nullclines serves as a vital indicator of these shifts. Recognizing these intersections can unveil essential information about the potential transition from stability to instability, or vice versa, providing insights into the dynamic behavior of the system.

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Most popular questions from this chapter

Consider the vector field given in polar coordinates by \(\dot{r}=r-r^{2}, \dot{\theta}=1\). a) Compute the Poincar茅 map from \(S\) to itself, where \(S\) is the positive \(x\)-axis. b) Show that the system has a unique periodic orbit and classify its stability. c) Find the characteristic multiplier for the periodic orbit.

(Globally coupled oscillators) Consider the following system of \(N\) identical oscillators; $$ \dot{\theta}_{i}=f\left(\theta_{i}\right)+\frac{K}{N} \sum_{j=1}^{N} f\left(\theta_{j}\right), \text { for } i=1, \ldots, N $$ where \(K>0\) and \(f(\theta)\) is smooth and \(2 \pi\)-periodic. Assume that \(f(\theta)>0\) for all \(\theta\) so that the in-phase solution is periodic. By calculating the linearized Poincar茅 map as in Example 8.7.4, show that all the characteristic multipliers equal \(+1\). Thus the neutral stability found in Example 8.7.4 holds for a broader class of oscillator arrays. In particular, the reversibility of the system is not essential. This example is from Tsang et al. (199I).

(Interacting bar magnets) Consider the system $$ \begin{aligned} &\dot{\theta}_{1}=K \sin \left(\theta_{1}-\theta_{2}\right)-\sin \theta_{1} \\\ &\dot{\theta}_{2}=K \sin \left(\theta_{2}-\theta_{1}\right)-\sin \theta_{2} \end{aligned} $$ where \(K \geq 0\). For a rough physical interpretation, suppose that two bar magnets are confined to a plane, but are free to rotate about a common pin joint, as shown in Figure \(1 .\) Let \(\theta_{1}, \theta_{2}\) denote the angular orientations of the north poles of the magnets. Then the term \(K \sin \left(\theta_{2}-\theta_{1}\right)\) represents a repulsive force that tries to keep the two north poles \(180^{\circ}\) apart. This repulsion is opposed by the \(\sin \theta\) terms, which model external magnets that pull the north poles of both bar magnets to the east. If the inertia of the magnets is negligible compared to viscous damping, then the equations above are a decent approximation to the true dynamics. a) Find and classify all the fixed points of the system. b) Show that a bifurcation occurs at \(K=\frac{1}{2} .\) What type of bifurcation is it? (Hint: Recall that \(\sin (a-b)=\cos b \sin a-\sin b \cos a .)\) c) Show that the system is a "gradient" system, in the sense that \(\dot{\theta}_{i}=-\partial V / \partial \theta_{1}\) for some potential function \(V\left(\theta_{1}, \theta_{2}\right)\), to be determined. d) Use part (c) to prove that the system has no periodic orbits. e) Sketch the phase portrait for \(0\frac{1}{2}\).

For each of the following systems, find the eigenvalues at the stable fixed point as a function of \(\mu\), and show that one of the eigenvalues tends to zero as \(\mu \rightarrow 0\). $$ \dot{x}=\mu-x^{2}, \quad \dot{y}=-y $$

(Logistic equation with periodically varying carrying capacity) Consider the logistic equation \(\dot{N}=r N(1-N / K(t))\), where the carrying capacity is positive, smooth, and \(T\)-periodic in \(t .\) a) Using a Poincar茅 map argument like that in the text, show that the system has at least one stable limit cycle of period \(T\), contained in the strip \(K_{\min } \leq N \leq K_{\max ^{\prime}}\) b) Is the cycle necessarily unique?
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