Dynamical system

A dynamical system combines a state space with a rule for evolution in time. A discrete rule sends a complete state $x_n$ to $x_{n+1}=F(x_n)$, while a continuous model can specify $\dot{x}=f(x)$. The state must include enough information for the chosen rule: a harmonic oscillator needs both position and velocity, because equal positions with opposite velocities have different immediate futures.

The same model can be studied through several questions. An equilibrium point stays fixed, a periodic orbit repeats, and a basin of attraction describes which initial states approach a selected long-term behavior. Changing a parameter can cause a bifurcation. These qualitative properties can be useful even when no explicit formula for every trajectory is available. Deterministic evolution also requires well-defined forward solutions; merely writing a differential equation does not establish existence or uniqueness of its initial value problem.

A dynamical observable is a function $\phi :X\to \mathbb{R}$ or $\mathbb{C}$ that assigns a measurable quantity to each state of a dynamical system. If the dynamics is $T:X\to X$, the observed sequence from state $x$ is $\phi (x),\phi (Tx),\phi (T^{2}x),\dots$. The state space may be high dimensional even when this record is scalar.

For a harmonic oscillator with state $(q,p)$, position $\phi (q,p)=q$ is one dynamical observable; energy is another. Position generally oscillates, while energy is constant along an ideal conservative trajectory. Composition $\phi \mapsto \phi \circ T$ describes evolution of the measured quantity instead of evolution of states. Its time average, $n^{-1}{\sum }_{j=0}^{n-1}\phi (T^{j}x)$, is the object compared with a space integral in ergodic theory.

For a map $T:X\to X$ and a given dynamical observable $f$, the additive cohomological equation asks for a function $u$ satisfyingWhen such $u$ exists, $f$ is a difference accumulated between consecutive states. Summing along a trajectory makes the intermediate terms cancel:Thus bounded $u$ forces these sums to remain bounded, and any periodic orbit of period $n$ must have sum zero. These are necessary conditions, not a universal existence theorem.

For the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$, try $f(x)=e^{2\pi ikx}$ and $u(x)=c{e}^{2\pi ikx}$. Then $c=(e^{2\pi ik\alpha }-1{)}^{-1}$ when the denominator is nonzero. If it vanishes, this mode is an obstruction; if it is very small, the coefficient is large. This explains how a simple equation leads to delicate regularity questions in dynamical systems.

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.

An additive cocycle over a dynamical system records a quantity accumulated along a flow. It satisfies $c(0,x)=0$ and $c(s+t,x)=c(t,x)+c(s,{\Phi }_{t}x)$: accumulation over two intervals equals accumulation over the first interval plus accumulation over the second interval from the evolved state.

For an dynamical observable $h$, the integral $c(t,x)={\int }_0^{t}h({\Phi }_{u}x)du$ is an example, as splitting the integral at $t$ proves the identity. A coboundary has the special form $c(t,x)=u({\Phi }_{t}x)-u(x)$. Matrix-valued multiplicative versions replace addition by an ordered product; the order matters when the matrices do not commute.

Synchronization of oscillators occurs when interacting or driven oscillators acquire a sustained relation between their phases or frequencies. Equal amplitudes are not required. In one simple phase model, a relative phase $\psi$ obeys $\dot{\psi }=\Delta \omega -K\sin \psi$, where $\Delta \omega$ is the uncoupled frequency mismatch and $K>0$ the coupling strength.

When $|\Delta \omega |<K$, a fixed relative phase exists; the branch with $\cos \psi >0$ is stable because the linearized restoring coefficient is $-K\cos \psi$. The Kuramoto model extends this competition between mismatch and coupling to a population of phase oscillators.^[ref]

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.

Repulsion in a dynamical system describes nearby trajectories moving away from an invariant state or set in forward time. For a scalar discrete dynamical system, a fixed-point multiplier with modulus greater than one gives local repulsion.

A forward-invariant set is a subset of a state space that contains every future state of each trajectory starting in it. For a discrete dynamical system with update map $F$, the condition isMathematical induction then gives $F^n(S)\subseteq S$ for every nonnegative integer $n$. For a flow map, the analogous condition is ${\Phi }_t(S)\subseteq S$ for every forward time for which the solution exists.

Forward invariance does not require equality or backward invariance. For $F(x)=x/2$, the interval $S=[0,1]$ is forward invariant because $F(S)=[0,1/2]\subset S$, but the inverse image of one is two, outside $S$. By contrast, the singleton $\{0\}$ is invariant in both directions.

This is useful before an exact solution is available. A forward-invariant positive interval prevents a population model from generating negative states and supplies an upper bound for all later iterates. The positive invariant interval for a cubic biological population map derives such a region by direct inequalities. Being bounded in a forward-invariant set does not alone imply convergence to one equilibrium point.

An invariant interval for a scalar map is an interval whose points remain in that interval under every forward iterate. Here invariance means forward invariance and does not assert surjectivity onto the interval.

For the discrete dynamical system $N_{n+1}=F(N_n)$ withevery positive starting value satisfying $F(N_0)>0$ has positive later iterates. The useful invariant interval is $(0,3/2)$.

First, for every $N>0$,because the quadratic has positive leading coefficient and discriminant $49-56<0$. Thus a positive first image lies below $3/2$.

Second, on $[0,3/2]$ the cubic product can be negative only for $1/2<N<1$. On that interval its magnitude is at most $(1/2)(1/2)(3/2)=3/8$, so $F(N)\ge 1/8$ throughout $[0,3/2]$. Together the bounds give $F((0,3/2))\subset (0,3/2)$. Mathematical induction then keeps all later values in that positive bounded interval.

The hypothesis on the first image is necessary: a positive starting population outside the invariant region need not have a positive image. The interval argument supplies positivity and boundedness; it does not by itself classify the eventual orbit.

A semiconjugacy from $f:X\to X$ to $g:Y\to Y$ is an intertwining map $h:X\to Y$:In a topological factor-map convention, $h$ is also continuous and onto. Authors using the term for an arbitrary intertwining map should state that weaker convention. If $h$ is a homeomorphism, the relation is a topological conjugacy.

The identity transports every forward orbit. Induction gives $h(f^n(x))=g^n(h(x))$, so iterating in the larger space and then projecting agrees with projecting first and then iterating. The map $h$ can merge different states, so the projected dynamics need not retain all information about the original orbit.

For example, let $f(\theta )=2\theta (\mathrm{mod}1)$ on the circle, $g(y)=4y(1-y)$ on $[0,1]$, and $h(\theta )={\sin }^2(\pi \theta )$. ThenThus the doubling map maps onto the parameter-four logistic map by a continuous semiconjugacy. Since $h(\theta )=h(1-\theta )$, it is not one-to-one and does not give a topological conjugacy by this map.

The forward orbit of a state is the collection of states reached as time advances. For a discrete dynamical system $F:X\to X$,For a flow map ${\Phi }_t$, the analogous set is $\{{\Phi }_t(x_0):t\ge 0\}$ over the times for which the solution exists. The initial state is included; its future accumulation points instead form the omega-limit set.

For example, $F(x)=x/2$ with $x_0=1$ gives $1,1/2,1/4,\dots$. Zero is approached but is not one of those iterates, so the forward orbit differs from its closure. Forward evolution also need not determine a unique past: $F(x)=x^2$ sends both $1$ and $-1$ to $1$. The distinction matters when discussing attraction or invariance for a map that cannot be inverted.

The box-ball system is a discrete cellular automaton with a row of boxes, each either empty or holding one ball. Consider finitely many balls and unlimited empty boxes to the right. During one update, process the original balls from left to right and move each exactly once to the nearest currently empty box on its right. The ordering and the instruction to move each ball only once are part of the evolution rule.

An isolated block of three balls in boxes $\{0,1,2\}$ moves to $\{3,4,5\}$ in one update. More generally, an isolated block of $k$ balls moves $k$ boxes per step, preserving its shape. Interacting patterns have a soliton interpretation: localized structures re-emerge with conserved identities, and the system has further conserved quantities.^[ref] This gives a particularly concrete setting in which a simple integer update rule produces organized wave-like motion.

Evolution is deterministic when a specified initial condition and evolution rule select a unique future orbit. For a discrete dynamical system, applying the same map to the same state gives the same next state. For an ordinary differential equation, uniqueness of the initial value problem is required; merely writing a differential equation does not guarantee it.

Determinism permits both asymptotic stability and sensitive dependence on initial conditions. It does not by itself imply reliable prediction from approximate measurements, and sensitivity does not introduce a random choice into the evolution rule.

The Lorenz system is the three-variable autonomous differential equationIt arose as a low-dimensional model of thermal convection. Edward Norton Lorenz used its numerical solutions in 1963 to study nonperiodic motion and sensitivity to the initial condition, with consequences for long-range weather forecasting.^[ref]

The equilibrium points follow directly from the equations. For $\sigma >0$ and $\beta >0$, the first gives $y=x$ and the third gives $z=x^2/\beta$. The second then becomes $x(\rho -1-x^2/\beta )=0$. Thus the origin always exists, while for $\rho >1$ there are two additional equilibriawith matching signs in the first two coordinates. The symmetry $(x,y,z)\mapsto (-x,-y,z)$ exchanges them.

The Equilibrium bifurcations of the Lorenz system determine how their stability changes. The separate instantaneous Lorenz error growth at a zero-y crossing calculation concerns a perturbation at one instant; it should not be confused with the long-term Lyapunov exponent.

For positive $\sigma ,b,r$, the origin of the Lorenz system loses stability at $r=1$ and two equilibria $(\pm \sqrt{b(r-1)},\pm \sqrt{b(r-1)},r-1)$ emerge stably. Center-manifold projection gives $\dot{q}=\sigma (r-1)q/(\sigma +1)-\sigma q^3/[b(\sigma +1)]+\cdots$, proving a supercritical pitchfork. The nonzero-branch characteristic polynomial is ${\lambda }^3+(\sigma +b+1){\lambda }^2+b(\sigma +r)\lambda +2\sigma b(r-1)$. If $\sigma \le b+1$ these equilibria remain stable for all $r>1$. Otherwise their transverse Hopf crossing occurs at the displayed parameter, with frequency squared $b(\sigma +r_H)$. Its cubic Hopf coefficient is positive, so this Hopf is subcritical, producing an unstable small periodic orbit on the stable-equilibrium side. A zero eigenvalue detects the pitchfork, while the Hopf condition uses the Routh-Hurwitz equality; neither alone determines nonlinear criticality without the corresponding normal-form calculation.

Consider the Lorenz system with $\sigma >0$ and $\beta >0$, at a state where $y=0$. An infinitesimal error $\eta$ evolves by $\dot{\eta }=J\eta$, where $J$ is the Jacobian matrix. Its squared Euclidean norm satisfiesAt that crossing, the skew couplings involving $x$ cancel, leavingThe planar block has negative trace and determinant $4\sigma -(\sigma +\rho -z{)}^2$. It has a positive eigenvalue exactly whenThus some error direction initially grows for either $z<\sigma +\rho -2\sqrt{\sigma }$ or $z>\sigma +\rho +2\sqrt{\sigma }$. The vertical direction initially contracts.

For $\sigma =10$, $\rho =28$, $z=26$ and $\beta =3$, the eigenvalues of $J+J^T$ are $4,-26,-6$. The most rapidly growing unit direction is $(1,2,0)/\sqrt{5}$, with instantaneous logarithmic norm growth $4/2=2$. The factor of two distinguishes squared-norm growth from norm growth.

This is a local criterion at the specified state. Later rotation of the perturbation direction can change its growth, so it does not prove a positive asymptotic Lyapunov exponent.

The Feigenbaum period-doubling map is the universal, even, quadratic-critical fixed map associated with repeated period-doubling bifurcations. One normalization isComposing twice and rescaling therefore reproduces the same graph. This self-similarity is the mechanism behind a common spatial scaling in otherwise different one-dimensional maps with a quadratic maximum.^[ref]

The quadratic maximum means $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Twice applying the chain rule at zero givesThe term involving $g^{\prime }(0{)}^2$ vanishes, and cancellation is legitimate because the maximum is nondegenerate. Thus the endpoint derivative is determined by the spatial scaling.

Writing $\gamma =-\beta >0$ makes the successive central intervals $[-{\gamma }^n,{\gamma }^n]$ shrink by the factor $\gamma$. Their iterates generate the Feigenbaum attractor; nonuniform interval lengths are encoded by the Feigenbaum geometric potential. The constant $\beta$ here describes spatial scaling, rather than the spacing of successive bifurcation parameter values.

The Feigenbaum geometric potential is a bounded negative symbolic potential $U$ encoding the lengths of nested intervals of the Feigenbaum attractor. In an appropriate binary coding,and changing only the distant tail of an address changes $U$ by an exponentially small amount. For odd interval labels, a sum along reversed address prefixes satisfieswith a bound independent of the depth and address. Even-label intervals have lengths comparable to neighboring odd-label intervals. These uniform estimates connect geometric lengths with thermodynamic formalism.^[ref][ref]

The bounded logarithmic error is important. For fixed real $t$, exponentiation givesAfter summing and taking $n^{-1}\log$, the error is at most $|t|C/n$, which vanishes. Thus geometric partition sums and symbolic partition sums give the same limiting topological pressure. The almost-additive partition-function limit supplies existence under the bounded-distortion hypotheses; bounded negative potentials give a strictly decreasing pressure.

This potential measures geometric contraction across partition levels. It is not an unbounded derivative potential evaluated at the critical point, where $g^{\prime }(0)=0$, and partition depth is distinct from elapsed orbit time.

Odd-address intervals lie in the first image of the central restrictive interval, away from the critical point. The derivative of the Feigenbaum map is bounded above and away from zero there, so the next even interval has comparable length by the mean value theorem. For the wraparound image, $g^{2^n}({\beta }^{n}x)={\beta }^{n}g(x)$ makes its length a fixed fraction $(1-\beta )/2$ of the central interval. Thus the odd and full partition length-power sums differ only by constant factors for each fixed real exponent, and have the same exponential growth rate.

For the normalized Feigenbaum map, put $\gamma =-g(1)$ and defineThe Feigenbaum attractor is the nested intersection $A={\bigcap }_n{\bigcup }_i{\Delta }_i^{(n)}$. Each level interval has two children, and the largest interval length tends to zero. Consequently every compatible infinite binary address selects one point of this Cantor set.^[ref]

At level $n$, the intervals are visited cyclically: the label changes from $i$ to $i+1$ modulo $2^n$. Expressing $i$ in binary therefore turns iteration on $A$ into the dyadic adding machine. One adds one with carries, rather than discarding the first symbol as in a full shift. For example, addresses written with the least significant digit first begin $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$.

There are $2^n$ geometric pieces at depth $n$, but those pieces are visited in a cycle. Counting them does not make the time evolution a two-symbol full shift. The Feigenbaum geometric potential uses this depth to study spatial contraction and Hausdorff dimension.

The dyadic adding machine is addition by one on the space of infinite binary digit sequences, interpreted with the least significant digit first. Carries propagate toward later digits: $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$. The all-ones sequence maps to the all-zeros sequence.

Equivalently, it is the compatible collection of cyclic maps $i\mapsto i+1(\mathrm{mod}{2}^n)$ for all $n$, or translation by one on the 2-adic integers. Every orbit visits every finite binary cylinder and is therefore dense. At the Feigenbaum attractor, this describes the order of visits to nested intervals; it is not the digit-discarding evolution of a full shift.

A piecewise differentiable interval map is uniformly expanding when a constant $\lambda >1$ bounds the derivative on every smooth branch:With a finite partition into monotone branches, a point's itinerary records which branch contains each iterate. A length-$n$ cylinder consists of points with the same first $n$ branch labels. Wherever this composition is smooth, the chain rule gives $|(T^n{)}^{\prime }|\ge {\lambda }^n$.

The mean value theorem then bounds the diameter of a cylinder by the length of the original interval times ${\lambda }^{-n}$. Inverse branches therefore determine starting points with exponentially improving spatial precision as more itinerary symbols are specified.

For the doubling map on $[0,1)$, the two branches have slope two and the level-$n$ cylinders are half-open intervals of length $2^{-n}$. Their labels are the first $n$ digits of the binary expansion. Endpoint conventions matter for unique coding, but not for the cylinder-length estimate.

Expansion describes the map's geometry. A symbolic potential separately specifies statistical weights, so expansion alone does not make every Gibbs measure absolutely continuous. The geometric potential of an expanding map uses the inverse derivative to select weights adapted to interval length.

For a differentiable uniformly expanding interval map, the geometric potential of an expanding map isThe exponential of its length-$n$ orbit sum is the reciprocal derivative magnitude $1/|(T^n{)}^{\prime }(x)|$. This weight is geometric: an inverse branch contracts interval lengths by that factor.

The corresponding Perron-Frobenius operator transports a density by summing contributions from inverse branches:For a finite smooth full-branch expanding map, standard bounded-distortion hypotheses give a positive regular fixed density $h$, normalized by $\int hdx=1$. The identity $\mathcal{L}h=h$ means that $h(x)dx$ is an invariant measure. The cylinder averages of an invariant density explain how the geometric cylinder weights recover that density.

For the doubling map, there are two inverse branches andThe constant density $h=1$ is fixed, giving Lebesgue measure. Choosing an unequal-symbol potential instead produces a different Gibbs measure; an arbitrary Gibbs measure need not be absolutely continuous. Thus the potential, rather than the symbolic alphabet alone, determines whether the measure agrees with geometric length.

The Perron-Frobenius operator transports a density under a map. For a piecewise smooth expanding interval map $T$, relative to Lebesgue measure,Each inverse branch contributes its incoming density times its length-contraction factor. A change of variables gives ${\int }_A\mathcal{L}fdx={\int }_{T^{-1}A}fdx$, proving that probability is transported correctly.

For the doubling map, $\mathcal{L}f(x)=[f(x/2)+f((x+1)/2)]/2$. Thus the constant density is fixed. More generally, a normalized nonnegative fixed density determines an absolutely continuous invariant measure; a constant density need not be fixed for an arbitrary expanding map.

Let a uniformly expanding interval map have cylinders ${\Delta }_w$ whose maximum length at level $n$ is at most $C{\lambda }^{-n}$, with $\lambda >1$. Suppose an invariant probability density $h$ is Lipschitz continuous with constant $L$ and satisfies $h\ge h_{\ast }>0$. Its cylinder probabilities are $\nu ([w])={\int }_{{\Delta }_w}hdx$. Define the piecewise constant logarithmic averageThe average of $h$ over ${\Delta }_w$ differs from $h(x)$ by at most $L\mathrm{diam}({\Delta }_w)$. Since the logarithm has derivative at most $1/h_{\ast }$ on this range,Thus the cylinder data recover the logarithm of the density uniformly at a geometric rate.

Conversely, take consistent positive cylinder probabilities and suppose these $g_n$ converge uniformly to a finite function $g$. Put $h=e^g$. If $\Vert g_n-g{\Vert }_{\infty }\le {\delta }_n\to 0$, integrating over every level-$n$ cylinder compares its $h$-mass with its assigned probability within factors $e^{\pm {\delta }_n}$. Subdivide any fixed cylinder into these finer cylinders and sum: consistency gives the same multiplicative bounds for the fixed cylinder. Letting $n\to \infty$ proves ${\int }_{{\Delta }_w}hdx=\nu ([w])$.

Shrinking cylinders determine the Borel probability measure, so the geometric image of the symbolic law is exactly $h(x)dx$. Use a consistent convention at branch endpoints; bounded limiting cylinder densities imply zero mass at individual endpoints. If the symbolic law is shift invariant, the coding relation transfers that invariance to the interval map. This converse is much stronger than inspecting a density ratio along just one itinerary.

Thermodynamic formalism studies weighted trajectories through partition functions, topological pressure and invariant measures. A symbolic potential $\varphi$ assigns an orbit segment the weight $e^{S_n\varphi }$, where $S_n\varphi$ is its accumulated potential. Summing these weights gives $Z_n(\varphi )$; the limiting logarithmic normalization per step is the pressure.

A one-symbol example shows the mechanism. On a two-symbol full shift, let $\varphi (0)=a$ and $\varphi (1)=b$. ThenNormalizing gives independent symbol probabilities $p_0=e^a/(e^a+e^b)$ and $p_1=e^b/(e^a+e^b)$. The associated Gibbs measure is therefore an orbit-weight law, not a uniform choice unless $a=b$.

The same construction can encode geometry. A geometric potential of an expanding map records inverse-branch contraction, while the Feigenbaum geometric potential encodes lengths in a nested partition. Their pressure functions organize spatial scaling and dimension calculations. In that second example, the partition depth is a geometric scale index and should not automatically be identified with elapsed orbit time.

This explains both the usefulness and the limitation of the statistical-mechanics analogy: changing the potential changes which trajectories or geometric pieces receive weight. An arbitrary Gibbs measure need not be absolutely continuous, even when its underlying interval map is expanding.

A Gibbs measure gives probabilities controlled by exponential weights from a potential. The precise convention must be specified: finite-volume conditional distributions and symbolic orbit-weight bounds are related formulations with different hypotheses.

For a finite-alphabet full shift, a standard symbolic Gibbs condition for a probability measure $\mu$ is that a constant $C\ge 1$ satisfyfor every length-$n$ word $w$ and every $x$ in its cylinder set $[w]$. Here $S_n\varphi (x)={\sum }_{j=0}^{n-1}\varphi ({\sigma }^{j}x)$, and $P(\varphi )$ is topological pressure. The same constant must work for every word and every length. Summable variations make the orbit-weight comparison within one cylinder uniform. In applications to equilibrium states, shift invariance is also required.

For a one-symbol potential $\varphi (x)=\log p_{x_0}$, where every $p_a>0$ and ${\sum }_a{p}_a=1$, the independent-symbol law hasThe bound is exact with $C=1$. Equal symbol probabilities and unequal symbol probabilities are both Gibbs laws, although their geometric images can be very different: an arbitrary Gibbs measure need not be absolutely continuous.

An arbitrary unnormalized list of one-sided weights need not give consistent conditional probabilities for every finite horizon. Normalization and compatibility are part of constructing the measure, rather than consequences of calling the weights Gibbs.

A Gibbs measure is determined by its potential, rather than by geometric volume alone. The doubling map $T(x)=2x(\mathrm{mod}1)$ gives a concrete counterexample. Away from dyadic rationals, write $x={\sum }_{j\ge 1}{\xi }_j{2}^{-j}$. Give the binary digits independent probabilities $\mathrm{Pr}({\xi }_j=1)=p$ and $\mathrm{Pr}({\xi }_j=0)=1-p$, with $0<p<1$.

A length-$n$ cylinder set with $k$ ones then has probability $p^k(1-p{)}^{n-k}$. This is the Gibbs measure for the one-symbol potential $\varphi (\xi )=\log p$ when ${\xi }_1=1$ and $\log (1-p)$ otherwise. The normalized topological pressure is zero because the length-$n$ weights sum to $(p+1-p{)}^n=1$.

Let $E_p$ be the set of numbers whose binary expansion has limiting frequency $p$ of the digit one. The strong law of large numbers gives ${\mu }_p(E_p)=1$. Under Lebesgue measure, binary digits are independent fair digits: every length-$n$ cylinder has length $2^{-n}$. The same strong law of large numbers therefore gives digit frequency $1/2$ almost surely. For $p\ne 1/2$, it follows that $\mathrm{Leb}(E_p)=0$.

Thus ${\mu }_p$ and Lebesgue measure are mutually singular measures. The ambiguity of dyadic expansions affects only a countable set with zero mass under both measures. The geometric choice is precisely $p=1/2$, when cylinder probabilities equal cylinder lengths.

Comparing ratios ${\mu }_p(C)/\mathrm{Leb}(C)$ on a few cylinders would be a weaker argument. Failure of uniform convergence to a positive bounded density does not exclude every possible integrable density; the full-measure set $E_p$ proves actual singularity.

For a finite full shift with alphabet $\mathcal{A}$ and a continuous potential $\varphi$ of summable variations, topological pressure is the limiting logarithmic weight per step:Here $w\eta$ concatenates a finite word and a fixed reference tail, and $\sigma$ is the left shift. Summable variations bound the effect of changing that tail by a constant independent of $n$. The almost-additive partition-function limit consequently proves existence and tail independence.

For a fixed potential $U$, the function $\gamma \mapsto P(\gamma U)$ is convex. At finite $n$, differentiating $n^{-1}\log Z_n(\gamma U)$ twice gives the variance of the orbit sum under its normalized weights, divided by $n$, which is nonnegative. Passing to the limit preserves convexity.

If $-B\le U\le -b<0$, then for $\delta >0$ each orbit weight is multiplied by a factor between $e^{-nB\delta }$ and $e^{-nb\delta }$. HencePressure is therefore strictly decreasing and Lipschitz continuous. If the alphabet has $m\ge 2$ symbols, $P(0)=\log m>0$ and $P(\gamma U)\to -\infty$ as $\gamma \to +\infty$. The intermediate value theorem then gives exactly one positive zero. For a one-symbol alphabet the zero is instead at $\gamma =0$; the alphabet-size qualification is essential.

For example, the constant potential $U=-c$, with $c>0$, has $Z_n(\gamma U)=m^n{e}^{-n\gamma c}$ and $P(\gamma U)=\log m-\gamma c$. Its zero is $(\log m)/c$, making the role of alphabet size explicit.

The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply Fekete's lemma to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the partition function. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.

A symbolic potential is a real-valued function on a symbolic sequence space, for example $U:{\mathcal{A}}^{\mathbb{N}}\to \mathbb{R}$ on a finite full shift. Its orbit sumweights a length-$n$ orbit segment in a partition function. Here $\sigma$ is the left shift. The potential can encode geometric contraction or a chosen statistical preference; it is not determined by the alphabet alone.

Its variation at depth $r$ isSummable variations mean ${\sum }_{r\ge 1}{\mathrm{var}}_{r}U<\infty$. If $x$ and $y$ agree on their first $n$ symbols, their shifted sequences agree for $n-j$ symbols at step $j$. ThereforeThe right side stays bounded independently of $n$. This uniform orbit-weight distortion is what allows a reference tail to be changed without changing topological pressure.

For a potential depending only on $x_0$, every variation with $r\ge 1$ is zero. The choice $U(x)=\log p_{x_0}$ generates independent-symbol weights, while a geometric potential of an expanding map instead describes inverse-branch contraction.

An integer matrix $A$ with determinant $\pm 1$ defines a toral automorphismon the real torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$. Integer entries make the map independent of the chosen representative. The inverse matrix also has integer entries, so the transformation is invertible. It preserves normalized Haar measure.

The map is a mixing measure-preserving transformation exactly when no eigenvalue of $A$ is a root of unity. This can be proved with characters of a real torus $e_k(x)=e^{2\pi ik\cdot x}$, since $e_k\circ T_A=e_{A^{T}k}$. Orthogonality givesIf no eigenvalue is a root of unity, a nonzero integer frequency cannot repeat: a repetition would give $((A^T{)}^m-I)k=0$ for some $m>0$. It therefore eventually leaves every finite set of frequencies. The displayed correlations prove mixing first for trigonometric polynomials, then for square-integrable functions by approximation.

Conversely, a root-of-unity eigenvalue makes $(A^T{)}^m-I$ singular for some $m$. As an integer matrix it has a nonzero rational kernel vector, which can be scaled to a nonzero integer vector $k$. The nonconstant character $e_k$ is then invariant under $T_A^m$, obstructing mixing. The cat map is a mixing example, while the identity matrix gives an invertible measure-preserving map that is not mixing.

The imperfect soft Duffing-van der Pol oscillatorcombines a softening restoring force, velocity-dependent damping and the symmetry-breaking term $2\epsilon v^2$. The equilibrium points are $v=0$ and $v=-\epsilon \pm \sqrt{{\epsilon }^2+\lambda }$, whenever the latter are real. The associated Jacobian matrix has trace $\kappa -v_{\ast }^2$ and determinant $\lambda -4\epsilon v_{\ast }-3v_{\ast }^2$.

At $\epsilon =0$, the equilibrium equation has the reflection symmetry $v\mapsto -v$. For $\epsilon \ne 0$, the branch near zero crosses the zero branch at $\lambda =0$, giving the generic transcritical bifurcation when $\kappa \ne 0$. The two nonzero roots meet at $\lambda =-{\epsilon }^2$, $v=-\epsilon$, giving a generic saddle-node bifurcation when $\kappa \ne {\epsilon }^2$. The excluded trace-zero cases have an additional linear degeneracy and require further analysis.

The mechanical energyhas the exact derivative $\dot{H}=(\kappa -v^2){\dot{v}}^2$. Thus the sign of energy input can change with displacement even at a fixed value of $\kappa$.

For $\lambda >0$, the two outer equilibria are saddle equilibria. Their potential barriers are generally unequal when $\epsilon \ne 0$, so a symmetric saddle connection can split into distinct directed connections under an imperfection. The weak-damping heteroclinic splitting of a tilted quartic oscillator calculates their leading parameter curves. Local equilibrium formulas alone do not guarantee a limit cycle or determine its global termination.

Pattern formation is the development of spatial structure in a system. A common mechanism is the loss of stability of a uniform state to disturbances at a preferred nonzero wavenumber. In Rayleigh-Bénard convection, heating a fluid layer from below can destabilize the conducting state and produce organized convection rolls. In a reaction-diffusion equation, reaction and transport can instead select a spatial concentration pattern.

A linear calculation separates the possible scales. Write a small disturbance as a Fourier mode $e^{\lambda (k)t+ikx}$. Positive real part of $\lambda (k)$ means that mode initially grows, and the fastest growing wavenumbers often set the first visible scale. The Swift–Hohenberg equation has $\lambda (k)=r-(1-k^2{)}^2$, so at small positive $r$ it amplifies a band near $k=\pm 1$.

Linear growth is only the onset calculation. A growing disturbance eventually becomes large enough for nonlinear terms to matter. Coupled amplitude equations can then describe competition between modes: one mode can suppress another, several can coexist, or the state can drift or oscillate. The resulting planform includes both the selected wavevectors and their relative amplitudes and phases.

A slowly varying phase modulation describes changes in a pattern's local position and orientation without immediately destroying its periodic structure. Its stability can differ from the stability of a uniform amplitude, as illustrated by the zigzag instability of parallel rolls.

A planform is the spatial arrangement of a pattern, especially its horizontal structure in a fluid layer. In a Fourier mode description it includes the active wavevectors, their relative amplitudes and their phases. Those data determine more than a list of preferred wavelengths.

For example, $w(x,y)=2A\cos (kx)$ describes parallel bands, with wavevectors $(\pm k,0)$. Adding an equal perpendicular pair gives $w(x,y)=2A[\cos (kx)+\cos (ky)]$, a square-symmetric planform. Unequal coefficients $2A\cos (kx)+2B\cos (ky)$ retain the two directions but generally lose the symmetry exchanging them.

A common translation symmetry changes the phases by $k_j\cdot s$ without changing the physical shape. Relative phase combinations that cannot be removed by a translation can distinguish different arrangements, as the translation invariants of Fourier-mode phases show. Thus choosing a planform identifies a symmetry type; the amplitude equations still have to determine whether that state exists and is stable.

A phase modulation changes the local position of a periodic pattern. If $w_0$ has period $2\pi$ in its argument, a transverse modulation can be writtenA constant $\varphi$ translates the entire pattern; a varying $\varphi$ bends its constant-phase lines. For the basic profile $w_0(\xi )=\cos \xi$, the local wavevector is $\nabla (x+\varphi )=(1,{\varphi }_y)$. The orientation changes at first order in the slope, while the local wavenumber is $\sqrt{1+{\varphi }_y^2}=1+{\varphi }_y^2/2+\cdots$.

In the method of multiple scales, a correction $C(\xi ,Y)$ depends both on its fast coordinate $\xi =x+\varphi (Y)$ and on its slow coordinate $Y$. Differentiating at fixed physical $x$ therefore uses the chain ruleFor instance, $C(\xi ){\varphi }_{YY}$ differentiates to $C^{\prime }(\xi ){\varphi }_Y{\varphi }_{YY}+C(\xi ){\varphi }_{YYY}$, not just the second term. The translation-covariant phase expansion calculates the second derivative needed in solvability conditions. Omitting the fast-coordinate dependence can change the nonlinear coefficient of the reduced amplitude equation while leaving its linear coefficients unchanged.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In the modela small Fourier mode has growth rate $\sigma (k)=-\lambda k^2-\gamma k^4$. Thus the straight pattern is linearly stable for $\lambda >0$ and unstable to sufficiently long transverse waves for $\lambda <0$.