Dynamical Systems: The Geometry Behind Differential Equations
From the nonlinear pendulum to stability and Poincaré’s qualitative dynamics
What does it mean to understand a differential equation?
Is finding an explicit solution always necessary? And when no simple formula exists, what can the geometry of the possible motions still tell us?
These questions lead to a different way of thinking about dynamics. Instead of following one solution through time, we can study how entire families of trajectories are arranged, how they behave near special states, and how their long-term evolution changes when the system is perturbed.
The nonlinear pendulum provides a simple place to begin. Its equation is easy to write down, but it already shows why solving an equation and understanding a dynamical system are not always the same thing.
A deceptively simple equation
A pendulum is one of the simplest systems in classical mechanics.
A mass hangs from a pivot and moves under the action of gravity. If θ denotes its angular displacement from the downward vertical and ℓ the pendulum length, its motion is described by
For small θ,
so the equation becomes
We recognize the harmonic oscillator, with its well-known sinusoidal solution
In this regime, the motion can be written explicitly and is easy to interpret
However, once the small-angle approximation is removed, the sin θ term makes the system nonlinear and the equation is no longer solved by elementary trigonometric functions. Its exact solution instead involves elliptic functions, a class of special functions, so the explicit time evolution is less straightforward to interpret.
The nonlinear pendulum can behave in several qualitatively different ways. It may oscillate back and forth around its downward equilibrium, or, with sufficient initial velocity, rotate continuously around the pivot. The downward and upward equilibrium positions have different stability properties, and a special trajectory separates oscillatory motion from continuous rotation.
By the late nineteenth century, differential equations had become fundamental to mechanics and mathematical physics, but many nonlinear equations resisted simple analytical solutions, and their possible behaviors were not always apparent from the equations themselves. The difficulty was particularly important in celestial mechanics, where questions about long-term motion and stability remained even when explicit solutions were unavailable.
Henri Poincaré began developing a qualitative theory of differential equations in the 1880s, asking what could be learned about solutions even when they could not be written explicitly. Rather than following only individual solutions, he studied the geometry formed by entire families of trajectories.
If we take the example of the pendulum, the idea is to describe its instantaneous state not by its angle θ alone, but by a point in a plane whose coordinates are its angle and angular velocity,
As the pendulum evolves, this point traces a curve. Different initial conditions generate different curves, and taken together these trajectories describe the geometry of the motion.
So, instead of asking only for the solution θ(t), we can ask how all possible trajectories are arranged in state space. This change of viewpoint is one of the central ideas of dynamical systems theory.
It allows us to identify equilibrium points, determine their stability, distinguish oscillation from rotation, and find the trajectories that separate different regimes of motion, often without requiring an explicit formula for the full time evolution.
The pendulum is simple enough for these ideas to be seen directly, but the same viewpoint extends much further, from nonlinear mechanics and celestial dynamics to stability theory and chaos.
From an Equation of Motion to a Dynamical System
The pendulum equation
is a second-order differential equation: determining its future motion requires both the initial angle and the initial angular velocity.
If we introduce
then the second-order equation can be written as a pair of first-order equations,
If the state of the pendulum is specified by the two-component vector
then its evolution can be written compactly as
The change of viewpoint, from the physical pendulum to a trajectory in state space, is summarized below.
This is a first-order autonomous dynamical system. It is first-order because only first derivatives appear, and autonomous because the right-hand side contains no explicit dependence on time. The evolution of the system at any instant is determined entirely by its current state.
The same form describes a much wider class of systems:
where the components of x specify the variables needed to describe the state of the system. A second-order equation for one variable therefore becomes a first-order system in two variables. More generally, higher-order equations and coupled equations can be rewritten in a similar form by enlarging the state vector.
At every point (θ, ω), the vector f(θ, ω) then specifies the instantaneous direction in which the state will move. Taken over the whole plane, these vectors define a vector field. Starting from an initial state (θ_0, ω_0), the evolving pendulum follows one curve through this field: its trajectory.
The collection of possible states forms the state space of the system. For mechanical systems, this is commonly called phase space. When the relevant phase space is two-dimensional, as for the pendulum considered here, it is called the phase plane. Each point represents a complete instantaneous state of the pendulum, while each trajectory represents one possible evolution in time.
💡 Note on phase space coordinates
In Hamiltonian mechanics, phase space uses
(θ, p_θ)withp_θ = mℓ²ω. For fixedmandℓ, this is just a constant rescaling of(θ, ω), so the qualitative phase-space geometry is unchanged.
The phase-plane representation is different from a conventional plot of θ(t). A time-series plot shows how one variable evolves for one chosen initial condition. The phase plane instead places all possible states in the same geometric space, allowing different kinds of motion to be compared directly.
For the pendulum, that geometry is organized into distinct dynamical regimes. Some trajectories form closed curves, others correspond to continuous rotations, and certain points remain fixed altogether.
To see why these different regimes appear, we can now examine the full set of trajectories in the phase plane.
Phase Plane and the Geometry of Motion
The phase plane makes it possible to examine all possible motions of the pendulum at once.
For the undamped pendulum, there is an additional quantity that organizes these trajectories: the total mechanical energy. Taking the lowest position of the pendulum, θ = 0, as the zero of gravitational potential energy, the total mechanical energy is
The first term is the kinetic energy and the second is the gravitational potential energy. In the absence of friction, this energy is conserved. Differentiating with respect to time gives
Using
the two terms cancel,
The pendulum thus remains on the same energy level throughout its motion. For a given value of E, its trajectory in the (θ, ω) plane must satisfy
Each trajectory is then a curve of constant energy. Plotting these trajectories for different initial conditions gives the phase portrait of the pendulum.
The phase portrait separates the motion into three qualitatively different regimes.
For sufficiently low energy, the pendulum cannot reach the vertically upward position. It moves back and forth around θ = 0, reversing direction whenever ω = 0. These bounded oscillations, usually called librations, appear in the phase plane as closed curves surrounding the downward equilibrium.
The boundary between libration and rotation occurs at a critical energy. At the vertically upward position,
and the energy is
At exactly this energy value, the trajectory forms the separatrix, the boundary dividing libration from rotation.
At higher energy, the pendulum has enough kinetic energy to pass over the top and continue moving in the same direction. This regime is called rotation; when the evolving variable is an angle, it is also commonly described as circulation. In the phase plane, rotational trajectories extend across successive values of θ rather than closing around the downward equilibrium.
During libration, ω changes sign at each turning point, whereas during rotation it retains the same sign. Since p_θ = mℓ²ω, the conjugate momentum behaves in the same way.
The qualitative behavior of the pendulum can thus be read directly from the geometry of its phase portrait, without solving explicitly for θ(t).
There is one further detail in this picture. The angle θ is periodic: values differing by 2π represent the same physical orientation. The periodicity of θ means that the pendulum's phase space can also be represented as a cylinder, obtained by identifying
Under this identification, librations remain closed loops around the stable equilibrium, while rotational trajectories wind continuously around the cylinder.
The phase portrait already distinguishes the main types of pendulum motion and shows how much information can be extracted from the geometry of trajectories alone. But mechanical motion and energy conservation are special features of this example. Before turning to fixed points and stability, it is useful to separate what belongs specifically to the pendulum from what applies more generally to dynamical systems.
Beyond the Pendulum: General Dynamical Systems
The pendulum provides a convenient example because its physical interpretation is immediate and its phase portrait already displays equilibria, closed trajectories, and a separatrix. But the same mathematical framework applies much more broadly.
A continuous-time autonomous dynamical system can be written in the general form
where
collects the variables required to specify the state of the system. The function f determines how that state changes from one instant to the next. In one dimension,
the state is described by a single variable. In two dimensions,
the state moves through a plane. More generally, an n-dimensional system evolves through an n-dimensional state space.
The state variables are simply the quantities needed to specify the system. They may represent interacting populations, chemical concentrations, electrical currents and voltages, or the coordinates and momenta of many bodies. They need not describe mechanical motion at all.
The pendulum has another property that is not shared by dynamical systems in general: in the absence of friction, it conserves mechanical energy. Energy conservation is specific to this example, not a defining property of dynamical systems. Other systems may dissipate energy, exchange it with their surroundings, or possess no meaningful mechanical energy at all.
Time can also enter the equations explicitly,
in which case the system is called non-autonomous. Here we will continue to focus mainly on autonomous systems, for which the current state alone determines the local direction of evolution.
Dynamical systems can also evolve in discrete steps rather than continuously, but the present discussion will remain with systems described by differential equations.
Despite these differences, the central geometric question remains the same: how does the system evolve through its state space?
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We can begin by asking the opposite question: are there states that do not evolve at all? These are the fixed points of the system, and their response to small perturbations leads directly to the concept of stability.
Fixed Points and Local Stability
For an autonomous dynamical system
a fixed point is a state x_* for which
If the system starts exactly at this point, it remains there forever, but this alone tells us nothing about what happens to nearby states. A fixed point is called stable if sufficiently small perturbations remain small. If nearby trajectories not only remain close but converge toward the fixed point, it is asymptotically stable. On the other hand, if arbitrarily small perturbations can grow and carry the system away, the fixed point is unstable.
This distinction was placed on a systematic mathematical basis by Aleksandr Lyapunov in his work on the stability of motion.
For nonlinear systems, the dynamics close to a fixed point can often be understood by examining a linear approximation. The idea is to start close to the equilibrium and ask how a small displacement evolves: does it grow, decay, or oscillate?
Suppose the state is displaced slightly from x_*,
where δx is small. Expanding the vector field around the fixed point gives
where
is the Jacobian matrix of the system. The Jacobian describes how the vector field changes locally when the state is displaced from the fixed point.
Since f(x_*) = 0, the local dynamics reduce to
So we can use this linearized system to study how small perturbations evolve. The relevant information is contained in the eigenvalues and eigenvectors of the Jacobian. The eigenvalues are obtained from the characteristic equation
If an eigenvalue has a negative real part, the associated perturbation decays; if it has a positive real part, the perturbation grows. A nonzero imaginary part produces oscillations. The associated eigenvectors determine the corresponding directions in state space.
For a two-dimensional system, these different eigenvalue combinations correspond to recognizable local geometries, summarized below.
The last case requires particular care. If the eigenvalues have zero real part, linearization by itself may not determine the stability of the full nonlinear system. Additional information about the nonlinear terms, conserved quantities, or other properties of the system may be required.
We can now apply this classification to the two equilibrium positions of the pendulum.
Fixed points require ω = 0 and sin θ = 0. Since the equilibrium conditions repeat whenever θ is shifted by 2π, there are two physically distinct configurations,
corresponding respectively to the downward and inverted positions.
The Jacobian of the pendulum system is
At the downward equilibrium, θ = 0, substituting cos 0 = 1 into the Jacobian gives the characteristic equation and resulting eigenvalues
The eigenvalues are imaginary, meaning the perturbations are oscillatory. In the phase portrait, nearby trajectories form closed curves around the equilibrium, which is called a center.
As noted above, purely imaginary eigenvalues do not by themselves establish nonlinear stability. For the pendulum, however, the conserved energy provides the additional information: trajectories sufficiently close to the downward equilibrium remain on nearby closed energy curves. The equilibrium is therefore stable, but not asymptotically stable.
At the inverted equilibrium, θ = π, the result changes to
One eigenvalue is negative and the other positive. Perturbations contract along one eigendirection and grow along the other, so the inverted equilibrium is an unstable saddle point.
This also explains a feature already visible in the phase portrait. The stable and unstable directions of the saddle extend into the surrounding phase plane, forming the trajectories associated with the separatrix between libration and continuous rotation.
For the ideal pendulum, however, the local analysis also shows a limitation: the stable downward equilibrium does not attract nearby trajectories. With no mechanism for losing energy, a displaced pendulum continues to oscillate indefinitely. Real systems behave differently when dissipation is present.
Dissipation and Attractors
The ideal pendulum conserves energy, so each trajectory remains on a constant-energy curve of the phase portrait. A real pendulum, however, loses energy through effects such as friction and air resistance.
A simple model adds a damping term to the equation of motion,
where γ > 0 measures the strength of the damping.
The mechanical energy is still
but it is no longer conserved. Differentiating with respect to time and using the damped equations of motion gives
meaning energy decreases whenever the pendulum is moving. In the phase plane, trajectories no longer remain on curves of constant energy. Instead, they cross successive energy levels as the oscillations lose amplitude.
The effect can also be understood from the local stability analysis developed in the previous section. At the downward equilibrium (θ, ω) = (0, 0), the Jacobian becomes
Its characteristic equation is
with eigenvalues
For weak damping,
the eigenvalues are complex conjugates with a negative real part. The center of the undamped pendulum is therefore replaced by a stable spiral: nearby trajectories spiral inward as the oscillations decay.
For sufficiently strong damping, the eigenvalues become real and negative, and the equilibrium instead behaves as a stable node. In both cases, the downward equilibrium is now asymptotically stable.
Without damping, nearby trajectories remain on closed energy curves around the center. With damping, they cross successive energy levels and approach the downward equilibrium as time increases, replacing the closed curves of the undamped system with inward trajectories.
The equilibrium therefore acts as a local attractor: states starting sufficiently close to it converge toward the same long-term state. Dissipation changes not only the energy of the system, but also the geometry of its flow, turning a center into an attractor.
Poincaré and the Qualitative View of Dynamics
Fixed points, linearization, and attractors describe important features of a dynamical system, particularly near an equilibrium. But they do not by themselves describe how trajectories are arranged across the full state space.
This broader geometric question was central to the work of Henri Poincaré in the late nineteenth century. Instead of asking only whether a differential equation could be solved explicitly, Poincaré studied the geometry and long-term evolution of its trajectories: where they could go, whether they returned, and how different kinds of motion were related.
One important possibility is periodic motion. The closed libration trajectories of the undamped pendulum provide a simple example.
A trajectory is periodic if there exists a period T > 0 such that
In state space, this translates into a closed orbit.
For systems with several degrees of freedom, however, the geometry of a full trajectory can become difficult to visualize. Poincaré introduced a way of reducing this complexity by recording only the points where a trajectory crosses a chosen lower-dimensional surface.
Let this surface be denoted by Σ. If a trajectory intersects it successively at
the evolution can be represented by a Poincaré return map
The continuous flow is thereby represented by a discrete sequence of section crossings.
This reduction preserves important information about the dynamics. If a periodic orbit returns to the same point on the section after each cycle, that point satisfies
The periodic orbit of the continuous system has become a fixed point of the return map. Questions about the stability of the orbit can then be studied through the evolution of nearby points under repeated applications of P.
The same idea extends beyond periodic motion. A complicated continuous flow may produce a much simpler geometric pattern when viewed only through its successive intersections with a section. The return map can therefore make recurring patterns and stability properties easier to identify than in the original differential equations.
Poincaré’s aim was not necessarily to obtain an explicit expression for x(t), but to determine the geometry of the motion: equilibria, periodic orbits, invariant trajectories, and the relationships between them.
Poincaré later applied this viewpoint to celestial mechanics, where the motion of several gravitating bodies provided precisely the kind of nonlinear system for which explicit solutions were generally unavailable. Return maps and related geometric ideas became part of the foundation of modern dynamical systems theory and, later, the mathematical study of chaotic motion.
From Equations to Geometry
We began with a question: to understand a dynamical system, is it always necessary to solve its equations of motion explicitly?
The nonlinear pendulum shows that the answer is no. Without writing its full solution in terms of elliptic functions, the phase portrait already distinguishes libration from rotation, identifies the separatrix between them, and shows that the downward equilibrium is stable while the inverted equilibrium is unstable.
Local analysis adds another level of understanding. Linearization around a fixed point, together with the eigenvalues and eigenvectors of the Jacobian, determines whether nearby trajectories remain close, move away, or converge toward the equilibrium. Adding dissipation changes this geometry again, replacing the closed trajectories of the ideal pendulum by trajectories that approach an attractor.
Poincaré’s qualitative approach extends the same idea beyond the pendulum. Periodic orbits, return maps, and trajectories across state space can be studied without requiring an explicit expression for every possible motion.
Explicit solutions remain valuable when they are available. But dynamical systems theory shows that they are not the only way to understand a nonlinear system.
The Python code used to generate all figures and diagrams in this article is open-source and available in the GitHub companion repository.
References & Further Reading
- Poincaré, H. (1881–1886). Mémoire sur les courbes définies par une équation différentielle / Sur les courbes définies par les équations différentielles. Journal de Mathématiques Pures et Appliquées. Four parts: Part I (1881), Part II (1882), Part III (1885), Part IV (1886).
- Poincaré, H. (1890). Sur le problème des trois corps et les équations de la dynamique. Acta Mathematica, 13, 1–270. Henri Poincaré Papers.
- Poincaré, H. (1892–1899). Les Méthodes Nouvelles de la Mécanique Céleste. Gauthier-Villars, Paris.
- Lyapunov, A. M. (1892; French trans. 1907). Problème général de la stabilité du mouvement. Annales de la Faculté des Sciences de Toulouse, 9, 203–474. Numdam.
- Birkhoff, G. D. (1927). Dynamical Systems. American Mathematical Society Colloquium Publications, Vol. 9.
- Guckenheimer, J., & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.
- Arnold, V. I. (1992). Ordinary Differential Equations. Springer-Verlag, Berlin Heidelberg. Springer.
- Perko, L. (2001). Differential Equations and Dynamical Systems. 3rd ed., Springer, Texts in Applied Mathematics, Vol. 7. 10.1007/978–1–4613–0003–8 · Springer.
- Hirsch, M. W., Smale, S., & Devaney, R. L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed., Academic Press. Elsevier.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. 2nd ed., Westview Press.
