Dynamical systems ($x_{n+1}=F(x_n),\dot{x}=f(x)$)

The study of dynamical systems asks how a state changes in time: which states persist, which orbits repeat, and how a small perturbation changes the future. A dynamical system combines a state space with an evolution rule. For a seasonal population, the state might be its abundance and one step one generation; for a harmonic oscillator, a state vector includes both position and velocity. Choosing a sufficient state is essential: position alone does not determine where an oscillator moves next.

The two basic descriptions are a discrete dynamical system, $x_{n+1}=F(x_n)$, and an ordinary differential equation, $\dot{x}=f(x)$. Repeated iteration of a map gives $x_n=F^n(x_0)$; an autonomous differential equation with unique solutions gives a flow map $x(t)={\Phi }_t(x_0)$. In either case, evolving for two successive durations agrees with evolving for their sum. The logistic map provides a particularly simple discrete dynamical system; the Lorenz system supplies a continuous-time example with a very different geometry.

Deterministic evolution says that the exact rule and exact initial condition specify one future. Predicting that future from measurements is a separate question. An instrument supplies a range of possible initial conditions, and a computation adds small errors. Near an asymptotically stable fixed point, nearby orbits converge and the long-term state can become easier to predict. With sensitive dependence on initial conditions, arbitrarily close starting states can eventually lead to a visible disagreement, even though each follows the same rule.

Much of the usefulness of dynamical systems comes from describing outcomes without writing every solution explicitly. Fixed points identify possible resting states; periodic orbits identify repeating behaviour; stability theory asks whether those behaviours survive small disturbances. A bifurcation diagram shows how changing a bifurcation parameter changes the possible outcomes. This is useful in population dynamics, where a stable abundance and an alternating abundance have different consequences, and in thermal convection, where a steady flow and a changing flow transport heat differently.

The beauty of the subject is that a small formula can organize a large variety of behaviour. The logistic map moves from an attracting fixed point to attracting periodic orbits and then to parameter values supporting sensitive motion. Its period-doubling cascade has a recognizable structure across many unimodal maps. Nevertheless, Devaney chaos, Glendinning chaos, and a positive Lyapunov exponent express different properties; a claim of chaos should specify its definition, its state space, and the parameter values under consideration.

Chaos describes sustained irregular dynamics with sensitive dependence in a deterministic system. The precise convention matters. Devaney chaos requires topological transitivity, dense periodic orbits and sensitive dependence on an invariant set. A common operational flow description asks for bounded nonperiodic long-term motion with sensitive dependence. These formulations are useful in different settings and are not universally equivalent. A positive largest Lyapunov exponent is a numerical diagnostic of exponential separation, not by itself a proof of every topological definition.

The softening restoring force and cubic velocity damping give the energy $E=(z^{\prime }{)}^2/2+z^2/2-z^4/4$ with $\dot{E}=-A(z^{\prime }{)}^2-(z^{\prime }{)}^4$. The origin and the two saddle points $z=\pm 1$ persist for all parameters. The origin undergoes a supercritical Hopf bifurcation at $A=0$. A heteroclinic cycle and a subsequent saddle-node bifurcation of periodic orbits are distinct global changes in its softening Rayleigh-Duffing phase portrait.

With $a=-A>0$, a stable cycle grows from the origin. At $a_h\simeq 0.33144619$, symmetric saddle connections form a repelling heteroclinic cycle, while the stable inner cycle persists. Just above it an unstable outer cycle coexists with the stable one; they merge at $a_f\simeq 0.33147066$ in a saddle-node bifurcation of periodic orbits. These approximate values follow from the half-return equations $H_a(p)=p$, with $H_a^{\prime }(p)=1$ at the fold, and saddle-manifold shooting at the connection. A coarse parameter scan can miss the narrow coexistence interval. The saddle eigenvalue ratio is below one at the connection, proving that the loop is repelling and preventing identification of the stable cycle's disappearance with that connection.

The Kuramoto model couples phase oscillators by sine differences. With all-to-all coupling, ${\dot{\theta }}_i={\omega }_i+Kr\sin (\psi -{\theta }_i)$, where $r{e}^{i\psi }$ is the Kuramoto order parameter. Natural-frequency diversity competes with coupling-induced phase coherence. In the symmetric stationary continuum branch, oscillators with $|\omega |\le Kr$ lock to the mean phase and the remainder drift.

The complex centroid of the oscillators' unit phase vectors has magnitude $r$ and argument $\psi$. A magnitude near one indicates phase coherence; zero indicates a balanced phase distribution, which need not be uniquely uniform at finite population size. Taking the imaginary part after multiplication by $e^{-i{\theta }_i}$ rewrites all-to-all sine coupling exactly in terms of these two collective variables.

For a symmetric stationary frequency distribution, locked oscillators contribute $r={\int }_{-Kr}^{Kr}\sqrt{1-{\omega }^2/(Kr{)}^2}g(\omega )d\omega$. The drifting oscillators' real contribution is zero by integrating a periodic logarithmic derivative, and imaginary contributions cancel by symmetry. Substituting $\omega =Kr\sin \theta$ and dividing by $r>0$ gives the equation. The incoherent solution must be retained before division. For finite positive continuous $g(0)$ the branch reaches it at $K_c=2/[\pi g(0)]$.

An autonomous differential equation in three variables. Its linear planar rotation couples to a nonlinear vertical variable through $xz$. Depending on parameters, its solutions can approach an equilibrium point, a periodic orbit, or a more complicated attractor. The specialization $b=a$ is used in the 2011 Oxford combined paper.

A chaotic attractor of a Rössler system for suitable parameter values. Local expansion near an unstable spiral combines with excursions in the third coordinate and global return; the existence of a local expanding direction by itself does not prove a chaotic attractor.

An invariant set approached by trajectories from a neighborhood in the state space of a dynamical system. An attracting equilibrium point and an attracting limit cycle are elementary examples. In more precise definitions, compactness and an appropriate stability or trapping condition are often also required.

Attraction in a dynamical system means convergence of nearby forward trajectories toward a specified invariant state or set. Attraction and Lyapunov stability are separate properties; together they give the usual local asymptotic stability of an equilibrium.

Under the geometric convention, a strange attractor is an attracting invariant set with irregular, typically fractal geometry rather than a smooth point, cycle or torus. Geometry and orbit instability are separate properties. A strange nonchaotic attractor can be geometrically strange without positive exponential separation. Some authors build chaos into the term instead, so that convention must be distinguished when using it.

A geometrically irregular attractor need not have chaotic exponential separation of nearby trajectories. Quasiperiodically forced systems provide primary examples in which strangeness and a positive largest Lyapunov exponent do not coincide. Consequently a fractal-looking numerical plot alone cannot establish chaos. ^[ref]

An equation specifying how a system changes with time, such as an ordinary differential equation for position and velocity or the Schrödinger equation for a wave function. Its solution depends on its specified initial conditions.

A modeling approach using stocks, flows and feedback to investigate coupled systems over time. Computer simulations can explore delayed responses and the consequences of different model assumptions.

Coauthor of The Limits to Growth by Donella H. Meadows and others.

Physicist and coauthor of The Limits to Growth by Donella H. Meadows and others.

Coauthor of The Limits to Growth by Donella H. Meadows and others.

Biophysicist and coauthor of The Limits to Growth by Donella H. Meadows and others.

A compact invariant set $K$ of a smooth invertible map $f$ is uniformly hyperbolic when its tangent spaces split continuously as $E^s\oplus E^u$, the derivative preserves this splitting, and constants $C>0$, $0<\rho <1$ satisfy $\Vert D{f}^n{v}_s\Vert \le C{\rho }^n\Vert v_s\Vert$ and $\Vert D{f}^{-n}{v}_u\Vert \le C{\rho }^n\Vert v_u\Vert$ for every $n\ge 0$. Thus the stable direction contracts forward and the unstable direction contracts backward. A Smale horseshoe is an example with nontrivial symbolic dynamics; such a set can contain infinitely many saddle periodic orbits without attracting an open neighborhood.

A Smale horseshoe is a compact hyperbolic invariant set of a smooth map with expanding and contracting directions and at least two strips that stretch across a common rectangle. Iterating the strips and their inverse images produces a Cantor-set product on which the dynamics are conjugate to a full shift. It supplies infinitely many saddle periodic orbits and chaotic itineraries; hyperbolicity alone does not make it an attracting invariant set. The geometric construction explains why phase-dependent reinjection near a Shilnikov bifurcation can create chaos even when the associated symmetric amplitude equations have no such mechanism.

An equivariant dynamical system has a vector field commuting with the action of a symmetry group. Its flow map preserves every subgroup's fixed-point subspace, because equivariance and uniqueness keep initial states fixed by that subgroup. Polynomial normal forms are constrained by the weights and parity of the group representation.

If a translation symmetry acts on complex Fourier modes by $A_j\mapsto e^{i{q}_j\cdot s}{A}_j$, a linear combination of their phases is invariant precisely when its coefficient vector lies in the kernel of the matrix of wavevectors. For four amplitudes whose wavevectors span a plane, this leaves two independent phase variables after quotienting translations. At zero amplitude, polar phases are undefined, so smooth centre manifold equations should first be written in Cartesian amplitudes.

An equivariant dynamical system can contain the monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in its $\ell$th amplitude equation only if its total wavevector equals the output wavevector. Its degree is ${\sum }_j(p_j+q_j)$. Thus an integer vector $d=p-q$ contributes at degree $N$ exactly when the wavevector equation holds, $\Vert d{\Vert }_1\le N$, and $N-\Vert d{\Vert }_1$ is even. The extra even degree is supplied by factors $|A_j{|}^2$. This finite integer test distinguishes regular saturation terms from phase-sensitive resonance terms.

Independent coordinate reflections require the $i$th component to be odd in $x_i$ and even in every other coordinate. Permutations force the same two cubic coefficients in all components. Absolute irreducibility forces a scalar linear coefficient, which can be normalized to the transverse bifurcation parameter. No quadratic terms occur. The axial branches with one, two or three equal nonzero absolute coordinates have leading squared amplitude $-\mu /[a+(m-1)b]$, provided that denominator is nonzero.

At a nondegenerate equivariant Hopf bifurcation, each complex axial isotropy subgroup of the spatial group combined with temporal phase gives a branch of periodic solutions with that spatiotemporal symmetry. Assumptions include an isolated semisimple imaginary critical pair of the required representation type, transverse parameter crossing, and a nonzero cubic coefficient on the fixed-point plane. The statement guarantees existence and symmetry type, not stability or which parameter side contains every branch.

A primary periodic branch whose phase advance is compensated by a spatial rotation. For the natural plane representation, the spatial pattern rotates rather than remaining on a fixed reflection axis. Opposite chiralities are exchanged by reflection.

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A primary periodic branch retaining a spatial reflection at every time. In the natural plane representation, each real-copy component oscillates along a fixed reflection axis. The fourfold dihedral case has distinct axial and diagonal standing-wave types.

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In Hopf coordinates where the rotation acts as $(i{q}_1,-i{q}_2)$, reflection swaps components, and temporal phase multiplies both equally, cubic equivariance permits precisely the three displayed monomials in the first component. Swapping indices gives the second equation, with the same complex coefficients. The rotating, axial standing, and diagonal standing branches are $q_2=0$, $q_2=q_1$, and $q_2=i{q}_1$. Their squared component amplitudes are $\mu /\mathrm{Re}a$, $\mu /\mathrm{Re}(a+b+c)$, and $\mu /\mathrm{Re}(a+b-c)$, respectively, when positive.

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Spatially complex coordinates on two real copies need not both have the same temporal phase weight. If their time-circle action is $(z_1,z_2)\mapsto (e^{-i\varphi }{z}_1,e^{i\varphi }{z}_2)$, conjugating the second coordinate gives a common Hopf phase. The linear terms then have the same imaginary frequency in $q$ coordinates but opposite imaginary frequencies in the original $z$ coordinates. Failing to make this conjugation produces amplitude equations inconsistent with the specified symmetries.

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A spatial group element combined with a time shift may preserve a periodic orbit even if the spatial element alone does not fix it at each instant. The temporal circle acts complex linearly on Hopf amplitudes. Complex axial isotropy subgroups identify the primary branches guaranteed by the equivariant Hopf theorem.

// Target: dynamical-systems.bigb

For the $D_3$ action $\rho (z_1,z_2)=({\zeta }^{-1}{z}_1,\zeta z_2)$, $m(z_1,z_2)=(z_2,z_1)$ and common phase multiplication, the fifth-order normal form has first component $z_1[\alpha +a|z_1{|}^2+b|z_2{|}^2+c|z_1{|}^4+d|z_1{|}^2|z_2{|}^2+e|z_2{|}^4]+f{\bar{z}}_1^2{z}_2^3$ and second component obtained by swapping indices. Here all coefficients may be complex. Its three primary symmetry types are a one-component rotating wave and the two standing-wave lines $z_1=z_2$ and $z_1=-z_2$, together with their group conjugates. The two standing amplitudes split at fifth order through $\mathrm{Re}f$.

If only reflection and common temporal phase remain, the linear complex matrix is $\left(\begin{array}{cc}\alpha & \eta \\ \eta & \alpha \end{array}\right)$. The symmetric and antisymmetric modes have eigenvalues $\alpha +\eta$ and $\alpha -\eta$, generically separating their Hopf thresholds. One-component rotating-wave axes are no longer invariant. Their hyperbolic finite-amplitude periodic solutions can persist as mixed-mode solutions, often through secondary symmetry-breaking bifurcations, while the double primary onset splits into two simple Hopf onsets. Coefficients determine the detailed secondary diagram and stability.

For an absolutely irreducible real symmetry representation, an equivariant steady-state bifurcation with a simple transverse crossing of the scalar linear eigenvalue generically has an equilibrium branch for each axial isotropy type. Restricting the vector field to its one-dimensional fixed-point subspace reduces the existence problem to a scalar bifurcation equation. The lemma guarantees these branches; it does not exclude branches with smaller isotropy or higher-dimensional fixed-point subspaces.

The normalizer of an axial isotropy subgroup preserves its one-dimensional fixed-point subspace of a group action. If its quotient by the isotropy contains an element acting as minus one, the reduced scalar vector field is odd, forcing a pitchfork bifurcation rather than a generic quadratic branch. For odd dihedral groups the reflection normalizer has no such quotient; an even-degree anisotropic term can distinguish the two half-axes at higher order.

// Target: algebra.bigb

For the natural plane action of the dihedral group, rotations constrain monomial weights and reflection makes coefficients real. The reflection axes support the bifurcating equilibria. For $n=3$, the quadratic anisotropic term gives three transcritical-like branches that are saddles in the full plane. For $n=4$, cubic isotropic and anisotropic terms determine two reflection-isotropy classes. For $n=5$, the cubic isotropic term sets the leading radial amplitude while quartic anisotropy separates stable and saddle angular classes.

// Target: dynamical-systems.bigb

Independent sign changes and interchange of two coordinates in the natural dihedral group action force the displayed cubic normal form. Apart from the origin, its nondegenerate small equilibria lie on coordinate axes or on the diagonals. An axial branch has squared amplitude $\mu /a$ and eigenvalues $-2\mu ,\mu (1-b/a)$. A diagonal branch has component squared amplitude $\mu /(a+b)$ and eigenvalues $-2\mu ,2\mu (b-a)/(a+b)$. Subtracting the two nonzero equilibrium equations gives $(a-b)(A^2-B^2)=0$, excluding other cubic branches when $a\ne b$.

For the four wavevectors $(2,1),(2,-1),(1,2),(1,-2)$ and their negatives, use invariant phases ${\chi }_1=2{\theta }_1-2{\theta }_2-{\theta }_3+{\theta }_4$, ${\chi }_2={\theta }_1+{\theta }_2-2{\theta }_3-2{\theta }_4$. Add to the cubic normal form of a dynamical system the two quintic terms $e{\overline{A}}_2{A}_3^2{A}_4^2$ and $f{\overline{A}}_1{A}_2^2{A}_3{\overline{A}}_4$ in the first equation, together with their square-symmetry partners; $e,f$ are real. For equal positive magnitudes, steadiness forces $\sin {\chi }_1=\sin {\chi }_2=0$ unless $e=f=0$. Equality of radial equations further gives $(e-f)(\cos {\chi }_1-\cos {\chi }_2)=0$. Consequently the generic phase types are $(0,0)$ and $(\pi ,\pi )$. When $e=f\ne 0$, the mixed type $(0,\pi )$, equivalent by square symmetry to $(\pi ,0)$, also exists. When both quintic coefficients vanish, every phase pair is allowed. Amplitude existence must still be checked from the remaining scalar equation.

A scalar mode that changes sign under a quarter-turn and is fixed by diagonal reflections couples quadratically through $AC,BC,A^2-B^2$. In the saturated three-mode equivariant dynamical system, a diagonal state $A=B=s$, $C=0$ has $s^2={\mu }_1/(1+\lambda )$ and $p=2(\lambda -1)s^2$. The symmetric perturbation decays at rate $2{\mu }_1$; the remaining stability matrix is the displayed block. With $K=2{\alpha }_1{\gamma }_1/(\lambda -1)>0$, its linear stability interval is $-K<{\mu }_2<-p$, provided $p<K$. The endpoints are a symmetry-breaking pitchfork bifurcation and a Hopf bifurcation, respectively, away from degeneracies.

A relative equilibrium is a solution that is stationary after undoing motion generated by a continuous symmetry. For complex amplitudes with a common rotation symmetry, constant moduli and constant relative phases can accompany a common phase evolving at constant frequency. Quotienting by the symmetry turns that solution into an ordinary equilibrium.

For identical point vortices on the plane, use $\omega ={\sum }_{a}d{x}_a\wedge d{y}_a$, ${\iota }_{X_H}\omega =dH$, and $L=\frac{1}{2}{\sum }_a(x_a^2+y_a^2)$. The generator $Y$ of simultaneous counterclockwise rotations is $-X_L$, and $L$ is its moment map with the convention ${\iota }_Y\omega =-dL$. In coordinates rotating with angular velocity $\Omega$, the Hamiltonian vector field is $X_H-\Omega Y=X_{H+\Omega L}$. Hence a collision-free configuration rigidly rotating at that velocity is exactly a critical point of the augmented Hamiltonian function $H+\Omega L$. For $H=-{\sum }_{a<b}\log |{\mathbf{r}}_a-{\mathbf{r}}_b{|}^2$, scaling all positions shows ${\sum }_a{\mathbf{r}}_a\cdot {\nabla }_{a}H=-N(N-1)$; a rotating relative equilibrium therefore obeys $2\Omega L=N(N-1)$. For $N\ge 2$ this implies positive $\Omega$ in the stated circulation convention. Reversing the contraction convention requires consistent changes to the Hamiltonian and rotation signs.

At a hyperbolic equilibrium, the unstable manifold is formed by nearby trajectories approaching that equilibrium as time tends to minus infinity. It is tangent to the span of eigenvectors whose eigenvalues have positive real part. Its dimension counts departing amplitudes; translation along an autonomous trajectory can remove one parameter when comparing orbit shapes. The stable manifold instead approaches the equilibrium forward in time.

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

For $x^{\prime \prime }+{\omega }^{2}x=\gamma \cos \theta$, $\dot{\theta }=\omega \ne 0$, use the section $\theta =0$ modulo $2\pi$ and $T=2\pi /|\omega |$. Exact resonance produces the secular term $\gamma t\sin (\omega t)/(2\omega )$, giving the displayed return translation. Nonzero forcing makes every section orbit unbounded. At zero forcing the map is the identity. The assumption $\omega \ne 0$ is essential: at zero frequency no transverse phase section exists and the motion is constant-acceleration or free-particle motion.

For $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, use entry $(x,h)$ and exit $(h,Y)$. If $\mu =k^2>0$, $x>k$, integration gives $Y=h[(h+k)(x-k)/((h-k)(x+k)){]}^{\lambda /(2k)}$. A smooth global return has $P(x)=\nu +cY+O(Y^2)$. If $\lambda >2k$, its derivative vanishes as $x\downarrow k$, and a small $\nu -k>0$ yields an attracting fixed point. If $\mu =-k^2<0$, the passage is $Y=h\exp [-(\lambda /k)(\arctan (h/k)-\arctan (x/k))]$. For small parameters the resulting global return is strongly contracting, with an attracting fixed point near $\nu$ on either side of zero.