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Nonlinear systems are mathematical models where the change in a system's state is dependent on non-linear interactions between variables. These systems are ubiquitous in the real world, capturing the complex dynamics of physical, biological, and social phenomena. In differential equations, a system is called nonlinear if at least one equation exhibits nonlinearity. Here, our system: \[ \begin{aligned} &\dot{x} = -y + xz^2 \ &\dot{y} = x + yz^2 \ &\dot{z} = -z(x^2 + y^2) \end{aligned} \] is nonlinear due to the terms involving products of variables, such as \( xz^2 \) and \( yz^2 \). Nonlinear systems can exhibit complex and rich behaviors, including chaos, periodic orbits, and bifurcations, which offer a deeper insight than linear systems can provide. When solving nonlinear systems, techniques often involve approximations to linear forms near equilibrium points or periodic orbits, such as with the Jacobian matrix and linearization.

The Jacobian matrix is a fundamental tool in examining nonlinear systems. It is a matrix of all first-order partial derivatives of a vector-valued function. For the system given by \( \begin{aligned} -y + xz^2,\ x + yz^2,\ -z(x^2 + y^2) \end{aligned} \), the Jacobian matrix \( J(x, y, z) \) is derived as: \[ J(x, y, z) = \begin{pmatrix} z^2 & -1 & 2xz \ 1 & z^2 & 2yz \ -2zx & -2zy & -(x^2 + y^2) \end{pmatrix} \] The Jacobian matrix provides insight into how perturbations in the state variables affect the system's behavior. By evaluating the Jacobian at particular points, such as periodic orbits or equilibrium, we can linearize the system around those points for simpler analysis. Evaluating at \( \gamma(t) = (\cos t, \sin t, 0)^T \): \[ J(\cos t, \sin t, 0) = \begin{pmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix} \] linearizes our system around the periodic orbit.

Characteristic exponents, also known as Lyapunov exponents, quantify the rate of divergence or convergence of nearby trajectories in a dynamical system. For a linearized system near an orbit \( \gamma(t) \, \) these exponents are given by the eigenvalues of the Jacobian matrix. In our example, the eigenvalues of \[ J(\cos t, \sin t, 0) = \begin{pmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix} \], are \( i, -i, -1 \). These eigenvalues indicate whether perturbations grow or decay exponentially over time. The real parts of the eigenvalues determine the stability of the system. If any eigenvalue has a positive real part, the corresponding direction is unstable, whereas a negative real part means stability. In our case, the characteristic exponents are \( i, -i, -1 \), revealing an oscillatory behavior (complex eigenvalues) and one decaying direction (real negative eigenvalue).

The fundamental matrix \( \Phi(t) \ \) is a matrix solution to the initial value problem \( \dot{\Phi}(t) = J(\gamma(t)) \Phi(t), \Phi(0) = I \). It describes the evolution of a linearized system's state over time. For our system linearized about \( \gamma(t) : \dot{x} = J(\gamma(t)) x \, \): \[ J(\cos t, \sin t, 0) = \begin{pmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix} \], the fundamental matrix can be formally expressed as: \[ \Phi(t) = \exp(J(\gamma(t)) t) = \begin{pmatrix} \cos t & -\sin t & 0 \ \sin t & \cos t & 0 \ 0 & 0 & e^{-t} \end{pmatrix} \]. This matrix encapsulates the effect of the Jacobian on perturbations along the periodic orbit.

Stable and unstable manifolds are critical in understanding the dynamics around fixed points and periodic orbits. The stable manifold of a periodic orbit \( \gamma(t) \, \) contains initial conditions that exponentially approach the orbit as time moves forward. Conversely, the unstable manifold consists of trajectories that exponentially diverge from the orbit. The dimensions of these manifolds correspond to the number of eigenvalues of the Jacobian matrix with negative and positive real parts, respectively. For \( J(\cos t, \sin t, 0) = \begin{pmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix} \), the dimensions are:

• Stable manifold: 1 (eigenvalue -1)
• Unstable manifold: 0 (no positive eigenvalues)
• Center manifold: 2 (eigenvalues \( i \) and \( -i \)

These manifolds help in visualizing the dynamical system's local behavior near the periodic orbit.