President Francois Hollande has called for vigilance after a weekend of attacks in France by men heard shouting Islamic slogans.
On Sunday a driver screaming "God is great" in Arabic ran down pedestrians in Dijon, injuring 11, two seriously.
It came a day after police shot dead a man in Tours after he attacked them with a knife, also shouting "God is great".
Mr Hollande urged the French on Monday not to panic.
The man who carried out Sunday's attack was arrested after targeting pedestrians in five different parts of the city in the space of half an hour.
He is said to be "apparently imbalanced" and to have spent time in a psychiatric hospital.
The lives of the two people seriously injured in Dijon are not said to be in danger.
Witnesses told police the driver, aged around 40, had also said he was "acting for the children of Palestine", an unnamed source close to the investigation told AFP news agency.
A spokesman for the interior ministry told French TV he believed the attacker had been acting alone.
The driver has been known to police for minor incidents dating back 20 years, he added.
The prosecutor in Dijon said the attacker had a long history of mental illness and the incident was not linked to terrorism.
Manuel Valls, the French Prime Minister, tweeted (in French) to express his "solidarity" with the victims.
In Saturday's incident, a man stabbed three police officers in the city of Tours before being shot dead.
Interior Minister Bernard Cazeneuve told French TV the man had been "very unstable".
Anti-terrorism investigators have opened an inquiry into the attack.
France has the largest number of Muslims in western Europe - estimated at between five and six million.
There have been a number of "lone wolf" attacks by Islamists in recent years.