Dynamical Systems#4: Linearization and Stability

This post introduces the crucial method of linearization for studying the qualitative behavior and stability of solutions near constant solutions (equilibria) of nonlinear autonomous dynamical systems.

Chapter 1: General Autonomous Systems and Equilibria

1.1. Representation of the System

We primarily focus on the general second-order autonomous system.

It is typically assumed that the functions f(x,y) and g(x,y) are “nice enough” (i.e., continuous and possessing continuous partial derivatives) so that every initial value problem has a unique solution.

1.2. Finding Equilibria

The most important step in understanding the solutions of such a nonlinear system is locating the constant solutions, which are called equilibria.

An equilibrium is a pair of numbers (a,b) such that, when x=a and y=b, the system is satisfied. Since an equilibrium is a constant solution, both derivatives must be zero: dx/dt=0 and dy/dt=0.

To find the equilibria, we solve the simultaneous algebraic system

Example 1 (Finding Equilibria):

Consider the system

We solve the corresponding algebraic system:

This leads to four possible cases, resulting in four equilibria

Chapter 2. Stability Concepts

Stability describes how solutions behave near an equilibrium

1. Attractor: An equilibrium c is an attractor if there is a region (a circle in the 2D case) around it such that any solution x(t) that starts inside that region approaches c as time t approaches infinity (t→∞). An attractor is an example of a stable equilibrium.

2. Stable: An equilibrium c is stable if every solution starting near c remains close to c.

3. Unstable: An equilibrium c is unstable if there is a region around it such that some solutions starting inside that region eventually leave and stay out of the region.

4. Repeller: If all solutions (excluding the equilibrium itself) starting inside the region eventually leave and stay out, c is a repeller. A repeller is an example of an unstable equilibrium.

Chapter 3: Stability of Linear Systems

For the simpler linear system

where x=0 is the equilibrium, stability is determined entirely by the eigenvalues of the coefficient matrix A.

• If all eigenvalues of A have a negative real part, x=0 is an attractor.

• If all eigenvalues of A have a positive real part, x=0 is a repeller.

• If A has eigenvalues with mixed real parts (positive and negative), or a zero eigenvalue, or a pure imaginary eigenvalue, x=0 is neither an attractor nor a repeller.

Chapter 4: Linearization: The Hartman-Grobman Theorem

Since nonlinear systems are generally difficult to solve globally, we study their local behavior near an equilibrium c by linearization.

4.1. The Linearization Matrix

The functions f(x,y) and g(x,y) are approximated near the equilibrium c=(a,b) using their Taylor series expansion. Since f(a,b)=0 and g(a,b)=0 (because c is an equilibrium), the approximation yields a linear system.

If we define a new variable y=x−c, the resulting linear approximation takes the form Dy=Ac​y.

The matrix Ac​, known as the linearization matrix (or Jacobian matrix), consists of the partial derivatives of f and g evaluated at the equilibrium (a,b)

4.2. The Hartman-Grobman Theorem

This theorem provides the link between the local dynamics of the nonlinear system and the linear approximation:

If the linearization matrix Ac​ has no zero or pure imaginary eigenvalues, then the phase portrait of the original nonlinear system near the equilibrium c is qualitatively similar (homeomorphic) to the phase portrait of the linearized system Dy=Ac​y.

In essence, if the eigenvalues are strictly stable (negative real part) or strictly unstable (positive real part), the linear system dictates the local stability of the nonlinear system

Chapter 5: Stability Analysis Examples

We apply the linearization method to analyze the stability of the equilibria found previously. The general Jacobian matrix for the functions

is

Example 2 (Stability)

The four equilibria are analyzed by substituting the (x,y) values into the Jacobian matrix Ac​ and finding the eigenvalues.

1. Equilibrium :

Eigenvalues: λ=1,1 (both positive). Conclusion: By Theorem, the linear system is a repeller. Since the eigenvalues are non-zero, the Hartman-Grobman Theorem applies, and is a repeller for the nonlinear system

2. Equilibrium :

Eigenvalues: λ=−1,−1 (both negative). Conclusion: The linear system is an attractor. is an attractor for the nonlinear system.

3. Equilibrium :

Eigenvalues: λ1​=−2,λ2​=−1 (both negative). Conclusion: The linear system is an attractor. is an attractor for the nonlinear system

4. Equilibrium :

Eigenvalues: λ1​=2/5 and λ2​=−1 (opposite signs). Conclusion: The equilibrium is unstable (specifically, a saddle point), as it has both positive and negative real parts among its eigenvalues.

The general phase portrait shows that trajectories in the first quadrant typically approach the attractors (1,0) or (0,1) as t→∞. The special trajectories that separate regions approaching different attractors are called separatrices.

Example # (Linearization of a Higher Order System)

We analyze the stability of the third-order system:

The equilibria are found by solving the algebraic system, yielding (0, 0, 0) and (-2,-2,0).

The general Jacobian matrix is

1. Equilibrium (0, 0, 0):

Eigenvalues: λ1​=−4,λ2​=−2,λ3​=1. Conclusion: Since one eigenvalue is positive (λ3​=1), the equilibrium (0,0,0) is unstable

2. Equilibrium (-2,-2,0):

Eigenvalues: λ1​=−4,λ2​=−2,λ3​=−1. Conclusion: Since all eigenvalues are negative, the equilibrium (−2,−2,0) is an attractor.

Chapter #5: Example

Find all the fixed point and stability of following non linear differential equations.

Let’s first find fixed point at f=0 and g=0. This give following.

Now let’s determin the stability of these equations using Jacobian matrix.

Let’s try all fixed points one by one.

Reference

Enjoy learning !!!