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A saddle point of a scalar function is a stationary point with nearby values both greater and less than its value. An indefinite Hessian matrix proves this by choosing directions with opposite signs of the quadratic term. A degenerate Hessian matrix may require higher-order terms: $x^2-y^4$ has a saddle point of a scalar function at the origin despite a zero quadratic coefficient in the $y$ direction. This is distinct from a saddle equilibrium of a dynamical system.

Stability theory studies the response of a dynamical system to small changes of its state. Linear stability of a planar equilibrium describes the response near equilibria using the linearized evolution.

State estimation infers an unobserved state of a dynamical system from its observations and a model of its evolution. An observed position can help estimate an unobserved velocity; repeated noisy observations can help distinguish changes in the state from measurement noise. The result should describe both an estimate and its uncertainty. A Kalman filter does this exactly for a linear Gaussian state-space model, maintaining a conditional expectation and conditional covariance.

The time at which information is available matters. Causal state filtering estimates the present state from observations available now. State prediction estimates a future state from those same observations. Later observations can improve an estimate of a past state, but that is a different information set from the one a live controller had. Kalman explicitly distinguishes these estimation tasks in A New Approach to Linear Filtering and Prediction Problems (1960), section Optimal Estimates.

State prediction estimates a future state of a dynamical system using observations already available. A state-space model supplies the transition law. Unlike causal state filtering at the next time, the prediction has not yet incorporated that next observation; a Kalman prediction step therefore includes uncertainty from new process noise.

The state space is the set of complete states admitted by a dynamical system. A point in this space specifies the information needed to apply the evolution rule. For one particle moving along a line, a mechanical state can be $(x,v)$: the same position with two different velocities represents two distinct states.

Constraints can restrict the admissible set, and changing the model changes what counts as a state. A fixed-length pendulum's angle is periodic, while its angular velocity is a separate coordinate. An orbit traces evolution through this space, and an equilibrium point is one state whose evolution remains fixed. Plotting an observed coordinate against time is therefore different from plotting a phase portrait. The latter can reveal loops, attraction and separatrices that a single-coordinate trace conceals.

A state variable is a coordinate used to describe the instantaneous condition of a dynamical system sufficiently completely to determine its future evolution under a specified model and input. For a simple pendulum, its angle and angular velocity form a state; angle alone generally does not, because the same position can be traversed in either direction.

A second-order ordinary differential equation $\ddot{q}=f(q,\dot{q},t)$ becomes a first-order system by setting $v=\dot{q}$: $\dot{q}=v$ and $\dot{v}=f(q,v,t)$. The choice of state variables therefore reflects the memory required by the model, and differs from merely listing measured outputs.

A storage function assigns a nonnegative quantity to the state of a dynamical system so that its increase is bounded by the integrated supply rate. In dissipativity, it must satisfyfor every allowed history and interval. A storage function may represent physical energy, but it can also certify an abstract input–output bound. Jan C. Willems's theory relates possible storage functions to recoverable supply, rather than selecting an arbitrary positive expression.^[ref]

For a differentiable storage function and state equation $\dot{x}=f(x,u)$, the pointwise inequalityis sufficient: integrating it along a solution proves the interval statement. Positivity alone is insufficient; the derivative must be checked against the specified supply rate.

An original dimensionless example makes this check exact. For $\dot{x}=-x+u$, output $y=x$, and supply rate $w=uy=ux$, try $S_a(x)=a{x}^2/2$ with $a>0$. Its derivative isThe difference from supply is ${\dot{S}}_a-ux=-a{x}^2+(a-1)xu$. Requiring it to be nonpositive for every real $u$ and fixed nonzero $x$ forces $a=1$. That value works because ${\dot{S}}_1=ux-x^2\le ux$. Thus even within this simple positive quadratic family, the supply law fixes the admissible scale.

A storage function also limits extraction. Rearranging the interval inequality givesStarting in a given state, one cannot withdraw more net supply than the initial storage certificate permits. A chosen storage function need not make that upper bound attainable.

When unforced operation makes the supply rate zero, storage cannot increase. This resembles a Lyapunov function, but nonincrease alone does not prove asymptotic stability: the function may fail to distinguish some states, or its derivative may vanish along persistent motion. The storage balance and a convergence proof are separate claims.

In a subcritical Hopf bifurcation, the small periodic orbit repels within the critical centre manifold and lies on the side where the equilibrium point attracts within that manifold. With the critical eigenvalue real part increasing through zero, the positive cubic sign gives the exact radial Hopf normal formFor $\mu <0$, the nonzero radius is $r_{\ast }=\sqrt{-\mu }$. Radii below $r_{\ast }$ shrink to zero, whereas radii above it grow. The radial derivative at the cycle is $\mu +3r_{\ast }^2=-2\mu >0$, proving that the periodic orbit repels. For example, $\mu =-0.04$ gives an unstable circle of radius $0.2$: a radius of $0.1$ decays while one of $0.3$ grows. The unstable circle shrinks into zero as $\mu \uparrow 0$; at $\mu =0$, $\dot{r}=r^3$ already drives every small positive radius outward.

For $\mu >0$ there is no nonzero constant radius in this cubic model, and zero repels. In a general dynamical system, this says only that the nearby orbits leave the local neighbourhood. It does not guarantee a distant attracting limit cycle, a jump to a particular amplitude, or hysteresis. Additional nonlinear or global information is needed; the quintic radial Hopf equation illustrates one possible stabilizing extension. Reversing the parameter orientation reverses which numerical side contains the branch, so the defining distinction between supercritical Hopf bifurcation and subcritical Hopf bifurcation is their reduced stability and their relation to the equilibrium point, rather than the bare sign of an arbitrarily named parameter.

A time average measures an dynamical observable along a single trajectory of a dynamical system. For a flow ${\Phi }_t$, define $A_{T}h(x)=T^{-1}{\int }_0^{T}h({\Phi }_{t}x)dt$. If the limit as $T\to \infty$ exists, it is the long-time average at initial state $x$. For a discrete map $T$, the corresponding expression is $n^{-1}{\sum }_{j=0}^{n-1}h(T^{j}x)$.

If $h({\Phi }_{t}x)=\cos t$, the average tends to zero although the observable never settles to a constant. The Birkhoff ergodic theorem supplies almost-everywhere existence for integrable observables under a probability-preserving map. Equality with a spatial average requires the appropriate ergodicity assumption; existence alone does not imply it.

Topological dynamics studies continuous maps and their orbits on topological spaces. For a compact metric space $X$ and a continuous map $T:X\to X$, the pair $(X,T)$ is a compact Hausdorff topological dynamical system.

A trapping region for a continuous-time dynamical system is a compact region whose boundary directs the vector field inward, so every forward trajectory that enters remains confined. For a smooth boundary $h(x)=0$ with interior $h<0$, a sufficient strict condition is $\nabla h\cdot f<0$ on the boundary. The chain rule then prevents a first outward crossing.

For $\dot{x}=-x$, $\dot{y}=-y$, the disk $x^2+y^2\le R^2$ is trapping because the orbital derivative of $x^2+y^2-R^2$ is $-2R^2$ on its boundary. A Lyapunov function often supplies such regions through its sublevel sets.

Forward invariance alone is weaker than strict inward trapping: a boundary can itself contain an invariant trajectory with zero normal velocity. The distinction matters when applying a theorem that requires strict boundary inequalities rather than confinement alone.

This real planar dynamical system has a saddle point at $(-1,0)$ and a repelling weak focus at $(1,0)$. The latter has purely imaginary linearized eigenvalues $\pm 2i$, so its linearization has closed elliptical orbits. Nonlinear terms nevertheless produce outward drift.

Set $X=2(x-1)$, $Y=y$, $\tau =2t$ and $r^2=X^2+Y^2$. The cubic expansion isThe linear rotation acts on scalar functions by $L=Y{\partial }_X-X{\partial }_Y$. For $H_2=r^2/2$, its cubic derivative is $X{Y}^2-3X^{2}Y/4-Y^3/2$. The homogeneous correction $H_3=7X^3/12+X{Y}^2/2+Y^3/3$ satisfies $L{H}_3$ equal to the negative of that derivative. The remaining quartic derivative of $H_2+H_3$ is $3X^2{Y}^2/2-X^{3}Y$. Withdirect differentiation givesfor all sufficiently small nonzero $r$. Meanwhile $H_2+H_3+H_4=r^2/2+O(r^3)>0$. In reversed time it is a strict Lyapunov function, proving asymptotic stability of the reversed system and hence repulsion forward in time. The angular speed stays close to $-1$, so the nearby nonzero trajectories spiral rather than approaching along a fixed direction. This explicit calculation illustrates why a centre in the linearization need not be a nonlinear centre.

Yakov Sinai contributed to the study of Feigenbaum constants through thermodynamic formalism, in joint work with E. B. Vul and Konstantin Khanin. Their 1984 paper connects universal period-doubling scaling with methods of statistical mechanics, illustrating how probabilistic and geometric descriptions of a dynamical system can reinforce each other.^[ref]